(university of antwerp, belgium)

Geometric functional analysis

Academic year 2017-18(autumn term)

Sonja Hohloch(University of Antwerp, Belgium)

Contents

Chapter 1. Motivation and introduction 11. What is geometric functional analysis? 12. Applications in topology: Morse theory 12.1. Motivation 1

Chapter 2. Classical theorems of functional analysis 31. Baire’s theorem and genericity 32. Uniform boundedness and the Banach-Steinhaus theorem 83. Open mapping, inverse mapping, and closed graph theorem 134. Seminorms and Frechet spaces 175. The Hahn-Banach extension and separation theorems 19

Chapter 3. Weak and weak* convergence and compactness 271. Strong, weak, and weak* convergence and reflexive spaces 272. Weak and weak* sequentially compactness of unit balls 293. Weak convergence of probability measures 32

Chapter 4. Fredholm theory 351. Definitions and examples 352. Adjoint operator on Banach spaces 363. Fredholm alternative 394. Persistence under perturbations 405. Inverse and implicite function theorems and Sard’s theorem 43

Chapter 5. Sobolev spaces and regularity theory 471. Definition and basic properties of Sobolev spaces 472. Weak and strong convergence in Sobolev spaces 513. Holder spaces 524. Regularity and compactness via embedding theorems 53

Chapter 6. Elliptic partial differential equations 571. Classical elliptic boundary value problem 572. Weak elliptic boundary value problem 583. Positive definite operators and the theorem of Lax-Milgram 614. Solving the weak elliptic Dirichlet boundary problem 63

i

ii CONTENTS

Appendix A. Appendix 671. Basic facts from topology 672. Basic facts concerning ordinary differential equations 673. Basic facts concerning differentiable manifolds 684. Basic facts from measure theory 695. Hausdorff maximality principle 69

Bibliography 71

Index 73

CHAPTER 1

Motivation and introduction

1. What is geometric functional analysis?

One of the fundamental tasks of mathematics is the study of maps from onespace to another space. One dimensional calculus focuses on functions

f : R→ R

and multivariable calculus on functions

f = ( f1, . . . , fn) : Rm → Rn.

Functional analysis studies maps between more general spaces, usuallyinfinite dimensional vector spaces with Banach or Hilbert structure. Suchmaps are often referred to as ‘operators’ or, if the target space is R or C, as‘functionals’. If one considers maps between infinite dimensional Banachand Hilbert manifolds instead of Banach and Hilbert vector spaces one oftenspeaks of global analysis rather than functions analysis. Geometric func-tional analysis aims at presenting the theorems and methods of ‘classical’functional analysis and showing their applications in geometry and topol-ogy.

We will try now to get an impression of examples, applications, and strengthof geometric functional analysis in topology and dynamical systems. Sinceeach of the two following examples is in fact an important theory on its own,we will only give a rough outline of the constructions and phenomena. Laterin the course, we will sometimes come back and study specific aspects indetail.

2. Applications in topology: Morse theory

2.1. Motivation. For standard notions from differential geometry like(Riemannian) manifold, (Riemannian) metric, and Hessian etc., we refer toDefinition A.7 in the appendix.

Definition 1.1. Let N be a smooth manifold and f ∈ C2(N,R). A pointx ∈ N is critical for f if the derivative vanishes in x, i.e., D f |x = 0. Wedenote the set of critical points by

Crit( f ) := x ∈ N | D f |x = 01

2 1. MOTIVATION AND INTRODUCTION

and call x ∈ Crit( f ) nondegenerate if the Hessian D2 f |x is nondegenerate.f is a Morse function if all points in Crit( f ) are nondegenerate.

Let us consider S2 := (x, y, z) ∈ R3 | x2 + y2 + z2 = 1 ⊂ R3 and its ‘heightfunction’ f : S2 → R given by f (x, y, z) := z. If we study the (change of)level sets f −1(r) for r ∈ R we observe

f −1(r) =

∅, r ∈ ] −∞,−1[,1 point, r = −1,1 circle, r ∈ ] − 1, 1[,1 point, r = 1,∅, r ∈ ]1,∞[,

i.e., the level sets undergo a drastic change precisely at the two criticalpoints of f given by the ‘south pole’ (0, 0,−1) and the ‘north pole’ (0, 0, 1).If we consider now the height function of a deformed sphere, we get evenmore possible shapes of level sets and transitions between them — for ex-ample ‘figure eight’ shaped curves when switching from one circle to twocircles and vice versa. But depending on the positioning of the deformedsphere in space we may also not notice the deformation at all by observingthe change of level sets!

Studying other examples makes us nevertheless notice that level sets (usu-ally) change at levels that contain critical points. This motivates the ques-tion:

Question 1.2. Can we somehow analyse the topology of a manifold N bystudying the critical points of a smooth function f : N → R? Or, vice versa,does the topology of the manifold enforce constraints on the number andpositioning of the critical points of f : N → R?

The general answer is yes and the specific answer what is infuenced bywhat is described by Morse theory. It is beyond the scope of this courseto present Morse theory in detail for which we refer to the literature[Audin & Damian], [Milnor], and [Schwarz], but we want to give at leasta rough sketch.

CHAPTER 2

Classical theorems of functional analysis

1. Baire’s theorem and genericity

In this section, we will study the following question: Given a topologicalspace X (cf. Definition A.1) and a subset V ⊆ X, we would like to measure‘how much’ of the space X is actually contained in V . In particular, we areinterested in subsets V that ‘fill almost everything’ of the space X, i.e., weare looking for a notion of ‘large’ and ‘small’ in the following sense:

• V ⊆ X is ‘large’⇔ X \ V is ‘small’.• Countable unions of large sets are large.• V ⊆ X ‘large’⇒ V , ∅.

If X carries the structure of a measure space with finite measure, i.e., if wehave a triple (X,A, µ) with σ-algebra A and finite measure µ, the naturalnotion of ‘large’ for V ∈ A would be µ(A) = µ(X) and for ‘small’ µ(V) = 0.

Is there also a meaningful notion if the space is not endowed with a mea-sure? The rough answer is yes and we now explain under which conditions.

First we recall without proof the following notion from general set topology.

Definition and Lemma 2.1. Let X be a topological space. V ⊆ X is dense inX if one of the following equivalent conditions is satisfied.

(1) V = X.(2) V ∩ O , ∅ for all open, nonempty sets O ⊆ X.

We are now interested in the following type of sets.

Definition and Lemma 2.2. Let X be a topological space. V ⊆ X is nowheredense in X if one of the following equivalent conditions is satisfied.

(1) Int(V) = ∅.(2) X \ V contains an open dense subset.(3) For all open, nonempty G ⊆ X, there exists an open, nonempty G ⊆ G

such that G ∩ V = ∅.

Proof. (1) ⇒ (2): The set U := X \ V ⊆ X \ V is open and dense byconstruction.

3

4 2. CLASSICAL THEOREMS OF FUNCTIONAL ANALYSIS

(2) ⇒ (3): Let U ⊆ X \ V be open and dense and let G ⊆ X be openand nonempty. Then the subset G := G ∩ U is open and G ∩ V = ∅ byconstruction.(3) ⇒ (1): Assume that Int(V) , ∅. By definition, Int(V) is open and wefind for all open G ⊆ Int(V) that G ∩ V , ∅. E

Obviously V ⊆ X is nowhere dense if and only if V is nowhere dense. Nowwe come to the main definitions of this paragraph.

Definition 2.3. Let X be a topological space and V ⊆ X.1) V is meager or of first (Baire) category if V is a countable union of

nowhere dense sets.2) V is nonmeager or of second (Baire) category if V is not of first (Baire)

category.

This property is named after the French mathematician Rene-Louis Baire(1871 – 1932). Intuitively, meager sets are ‘small’ and nonmeager sets‘large’.

Remark 2.4. Let X be a topological space.1) If V ⊆ X is meager and V ⊆ V then V is also meager.2) If, for all n ∈ N, the sets Vn ⊆ X are meager and Vn ⊆ Vn then

⋃n∈N Vn

is also meager.

In geometric applications, one is usually more interested in nonmeager setsthan in the meager ones. This prompts the following definition.

Definition 2.5. A topological space X is said to be Baire or a Baire spaceif the countable intersection of open and dense subsets of X is always dense.

There are in fact several equivalent ways to define Baire spaces.

Theorem 2.6. Let X be a topological space. Then the following statementsare equivalent.(1) X is Baire.(2) The countable union of closed, nowhere dense subsets of X has empty

interior.(3) Every meager subset of X has empty interior.(4) The complement of every meager subset of X is dense in X.(5) Every nonempty open subset of X is not meager in X.

Proof. We leave this proof as an exercise to the reader since it is ofpurely topological nature and does not lie within the scope of this course.The equivalence of items (1) and (2) can be found for example in[Chacon & Rafeiro & Vallejo, Th. 9.3].

Now we look for ‘natural’ Baire spaces.

1. BAIRE’S THEOREM AND GENERICITY 5

Theorem 2.7 (Baire). 1) Every complete metric space is Baire.2) Every locally compact Hausdorff space is Baire.

Proof. 1) Let (X, d) be a complete metric space and Vn ⊆ X open and densefor all n ∈ N. We have to show that V :=

⋂n∈N Vn is dense in X, i.e., for any

open subset B ⊆ X, we have B ∩ V , ∅.Since B is open and V1 is open and dense there exists a point x1 ∈ B ∩ V1.Since B ∩ V1 is open there exists 0 < ε1 < 1 such that B1 := B(x1, ε1) ⊆B∩ V1. Since B1 is open and V2 is open and dense there exists x2 ∈ B1 ∩ V2

and 0 < ε2 <12 such that B2 := B(x2, ε2) ⊆ B1∩V2. Iterating this procedure,

we get a nested sequence

B ⊃ B1 ⊃ B1 ⊃ B2 ⊃ B2 ⊃ · · · ⊃ Bn ⊃ Bn+1 ⊃ Bn+1 ⊃ . . .

whose radii satisfy 0 < εn <1n and therefore converge to zero. The sequence

of middle points (xn)n∈N is a Chauchy sequence since the nestedness impliesd(xk, xl) ≤ 2

N for all k, l ≥ N. Since (X, d) is complete x := limn→∞ xn existsand lies by construction in

⋂n∈N Bn which in turn lies in B ∩ V .

2) Left as an exercise to the reader. A proof may be found in [Lowen].

Example 2.8. Banach spaces are Baire.

Proof. The complete norm ‖ ‖ of the Banach space gives rise to a completemetric d via d(x, y) := ‖x − y‖.

In applications, often the following notion is used.

Definition 2.9. Let X be a topological space. V ⊆ X is called generic if Vis dense in X and can be written as the countable intersection of open sets.

Being ‘generic’ means intuitively to show ‘typical behaviour’ or to ‘behaveas most functions in a given situation behave’.

Now we want to know what kind of applications Theorem 2.7 (Baire) andthe notion ‘generic’ have in analysis and geometry. First we recall the fol-lowing phenomenon.

Remark 2.10. There exist continuous functions that are nowhere differen-tiable, for example

• The Weierstraß function W : R→ R given by

W(x) :=∞∑

n=0

bn cos(anπx)

with a ∈ N odd and b ∈ ]0, 1[ satisfying ab > 1 and 23 >

πab−1 .


• The Riemann type function

x 7→∞∑

n=0

1n2 sin(n2πx).

• Van der Waerden’s function

x 7→∞∑

n=0

infk∈Z |4nx − k|4n .

Now we want to answer the immediate question if being nowhere differ-entiable is a ‘typical’ or ‘rare’ property of continuous functions, i.e., is itgeneric or not?

Theorem 2.11. 1) The set f ∈ C0([0, 1],R) | f nowhere differentiable isgeneric in C0([0, 1],R).

2) The set f ∈ C0([0, 1],R) | f somehere differentiable is meager inC0([0, 1],R).

Proof. 1) Abbreviate I := [0, 1] and define for g ∈ C0(I,R) and x, y ∈ Isatisfying x + y ∈ I the function

Fgn(x, y) := |g(x + y) − g(x)| − n |y|

which measures how much the difference quotient in the definition of dif-ferentiability blows up. Moreover, define the subsets

Un := g ∈ C0(I,R) | ∀ x ∈ I ∃ y ∈ I : Fgn(x, y) > 0

whose intersection U :=⋂

n∈NUn consists of functions that are nowhere inI differentiable. We have to show that U is generic.Step 1: C0(I,R) equipped with d( f , g) := ‖ f − g‖∞ := supx∈I | f (x) − g(x)| isa complete metric space. Using Theorem 2.7 (Baire), it is enough to showthat all Un are open and dense in C0(I,R).Step 2: We show that Un is open, i.e., given g ∈ Un, we show that thereexists a small open ball around g that lies in Un.We notice that, for each g ∈ Un, there exists ε > 0 such that for all x ∈ Ithere is y = y(x) ∈ I with Fg

n(x, y) > ε. Note that this ε holds uniformly forall x ∈ I. The inequality Fg

n(x, y) > ε can be reformulated as

ε + n |y| < |g(x + y) − g(x)|

which gives rise to the following estimate for all h ∈ C0(I,R):

ε + n |y| < |g(x + y) − g(x)|≤ |g(x + y) − h(x + y)| + |h(x + y) − h(x)| + |h(x) − g(x)|

≤ ‖g − h‖∞ + Fhn(x, y) + n |y| + ‖g − h‖∞

1. BAIRE’S THEOREM AND GENERICITY 7

This implies the inequality 0 < Fhn(x, y) for all h with ‖g − h‖∞ < ε

2 . Thusthe whole open ball around g with radius ε

2 lies in Un, i.e., Un is open.Step 3: We have to show that Un is dense in C0(I,R), i.e., given g ∈ C0(I,R)and ε > 0, there exists h ∈ Un such that ‖g − h‖∞ < ε.It is easy to see that the set of piecewise linear continuous functions is densein C0(I,R). Therefore it is enough to work with a piecewise linear g fromnow on. Since I is compact there exists K > 0 such that g′ ≤ K whereverthe derivative g′ is defined. Consider the continuous function

ϕ : I → R, ϕ(x) := infk∈Z|x − k|

and choose m ∈ N such that εm − K > n. The function

h : I → R, h(x) := g(x) + εϕ(mx)

is continuous and, for 0 ≤ x < 1, we find

limy→0+

∣∣∣∣∣h(x + y) − h(x)y

∣∣∣∣∣ = limy→0+

∣∣∣∣∣g(x + y) + εϕ(m(x + y)) − g(x) − εϕ(mx)y

∣∣∣∣∣= lim

y→0+

∣∣∣∣∣g(x + y) − g(x)y

+ εϕ(mx + my) − ϕ(mx)

y

∣∣∣∣∣≥ |K − εm|> n

which implies h ∈ Un. Moreover, we estimate

‖g − h‖∞ = supx∈Iεϕ(mx) ≤

ε

2< ε.

2) follows immediately from 1).

For the construction of Morse theory, one makes use of Morse functions.Thus the question arises ‘how many’ Morse functions there are on a com-pact manifold.

Theorem 2.12. Let N be a smooth compact manifold without boundary.Then the set of Morse functions is generic in C∞(N,R).

Proof. This will be done either during the exercise sessions or homeworkor the interested reader may browse the standard literature on Morse theorylike [Audin & Damian], [Milnor], [Schwarz]. . .

Remark 2.13. Intuitively, properties are generic if they persist under smallperturbations. For example, properties like

• expression , 0 among continuous functions,• transversal intersections of smooth graphs

stay true after small perturbations and are thus (usually) generic.Conversely


• expression = 0 among continuous functions,• tangencies or tangential intersections of smooth graphs

usually become , 0 or transversal after perturbation, i.e., these are (usu-ally) nongeneric properties.

2. Uniform boundedness and the Banach-Steinhaus theorem

In this section, we study an extension to Banach spaces of the followingwell-known result:

Let f : K ⊂ R → R be continuous. Then, if K is compact, f is uniformlycontinuous, i.e., for all ε > 0 there is δ = δ(ε) > 0 such that for all x, y ∈ Kwith |x − y| < δ follows | f (x) − f (y)| < ε.

Let us introduce some notations before we investigate under which condi-tions ‘local, pointwise behaviour’ may lead to ‘controlled global behaviour’.

Definition 2.14. Let (X, ‖ ‖X) and (Y, ‖ ‖Y) be normed vector spaces over afield F and T : X → Y a map.

1) T is a linear operator if

T (λx + x) = λT (x) + T (x) ∀ x, x ∈ X, ∀ λ ∈ F.

2) A linear operator T is bounded if

∃ C > 0 : ‖T (x)‖Y ≤ C ‖x‖X ∀ x ∈ X.

3) The set of linear bounded operators from X to Y is denoted by B(X,Y).4) The operator norm of T ∈ B(X,Y) is given by

‖T‖ := sup‖T (x)‖Y | x ∈ X, ‖x‖X ≤ 1.

5) The space B(X,R) and B(X,C) are the real and complex dual space ofX. Sometimes they are denoted by X∗ or X′ in the literature.

We recall the following standard facts.

Remark 2.15. 1) For a linear operator, being bounded is equivalent to be-ing continuous which in turn is equivalent to being continuous at theorigin, cf. for instance [Hohloch1].

2) An equivalent definition of the operator norm is

‖T‖ = sup‖T (x)‖Y | x ∈ X, ‖x‖X = 1.

3) The operator norm depends on the choice of the norms ‖ ‖X and ‖ ‖Y .4) If Y is a Banach space then B(X,Y) is Banach, too. In particular, the

dual spaces B(X,R) and B(X,C) are Banach.

2. UNIFORM BOUNDEDNESS AND THE BANACH-STEINHAUS THEOREM 9

Now we want to study boundedness for families of linear operators. Givenan index set J, we abbreviate a family

T j ∈ B(X,Y) for all j ∈ J

by(T j) j∈J ∈ B(X,Y).

Definition 2.16. Let (X, ‖ ‖X) and (Y, ‖ ‖Y) be normed vector spaces, J anindex set, and (T j) j∈J ∈ B(X,Y).1) The family (T j) j∈J is pointwise bounded if

∀ x ∈ X ∃ Cx > 0 :∥∥∥T j(x)

∥∥∥Y≤ Cx ∀ j ∈ J.

2) The family (T j) j∈J is uniformly bounded if

∃ C > 0 :∥∥∥T j(x)

∥∥∥Y≤ C ∀ j ∈ J.

The following crucial theorem is named after Stefan Banach (Polish math-ematician, 1892 – 1945) and Hugo Dyonizy Steinhaus (Polish mathemati-cian, 1887 – 1972).

Theorem 2.17 (Banach-Steinhaus). Let (X, ‖ ‖X) be a Banach space and(Y, ‖ ‖Y) a normed vector space. Let J be an index set and (T j) j∈J, (S n)n∈N ∈

B(X,Y).1) If (T j) j∈J is pointwise bounded then (T j) j∈J is in fact uniformly bounded,

i.e., sup j∈J

∥∥∥T j

∥∥∥ < ∞.2) If sup j∈J

∥∥∥T j

∥∥∥ = ∞ thenx ∈ X

∣∣∣sup j∈J

∥∥∥T j

∥∥∥ < ∞is meager in X.

3) If limn→∞ S n(x) =: S (x) exist for all x ∈ X then (S n)n∈N is uniformlybounded and S ∈ B(X,Y) with ‖S ‖ ≤ supn∈N ‖S n‖.

Proof. 1) Step 1: We want to use Theorem 2.7 (Baire). The pointwiseboundedness of (T j) j∈J allows us to write

X =⋃k∈N

x ∈ X

∣∣∣ ∀ j ∈ J :∥∥∥T j(x)

∥∥∥Y≤ k

︸︷︷︸=:Fk

=⋃k∈N

⋂j∈J

x ∈ X

∣∣∣ ∥∥∥T j(x)∥∥∥

Y≤ k

︸︷︷︸=:Fk, j

.

Since T j is continuous, Fk, j = T−1j

(B(0, k)

)is closed for all j ∈ J and all

k ∈ N and so is Fk for all k ∈ N. Because of ∅ , Int(X) = Int(⋃

k∈N Fk), thereexists m ∈ N such that Int(Fm) , ∅ since we get otherwise a contradiction toX being a Baire space (see Theorem 2.6, item (2)). Since Int(Fm) is open andnonempty, there exists x ∈ Fm and r > 0 such that B(x, r) ⊆ Int(Fm) ⊂ Fm.By definition of Fm, we have

∥∥∥T j(y)∥∥∥

Y≤ m for all y ∈ B(x, r) and all j ∈ J.


Step 2: Given z ∈ B(0, 1) ⊆ X, the point z := x + r2z lies in B(x, r). We

rewrite z = 2r (z − x) and estimate∥∥∥T j(z)

∥∥∥Y≤

∥∥∥∥T j

(2r (z − x)

)∥∥∥∥Y

= 2r

∥∥∥T j(z) − T j(x)∥∥∥

Y≤ 2

r

(∥∥∥T j(z)∥∥∥ +

∥∥∥T j(x)∥∥∥

Y

)≤

4mr

for all j ∈ J. Since the upper bound 4mr does not depend on j, we obtain

supj∈J

∥∥∥T j

∥∥∥ = supj∈J

∥∥∥T j(z)∥∥∥

Y

∣∣∣ ‖z‖ ≤ 1≤

4mr< ∞,

i.e., (T j) j∈J is uniformly bounded.2) Using the notation from the proof or 1), we find

X =

x ∈ X

∣∣∣∣∣∣ supj∈J

∥∥∥T j(x)∥∥∥ = ∞

∪

x ∈ X

∣∣∣∣∣∣ supj∈J

∥∥∥T j(x)∥∥∥ < ∞

=

x ∈ X

∣∣∣∣∣∣ supj∈J

∥∥∥T j(x)∥∥∥ = ∞

∪

⋃k∈N

Fk

which is a countable union of closed sets. If now one of the Fk hadnonempty interior then, in analogy to the proof of 1), we would obtain that(T j) j∈J is uniformly bounded and in particular

x ∈ X

∣∣∣∣∣∣ supj∈J

∥∥∥T j(x)∥∥∥ = ∞

= ∅

in contradiction to our assumption. Thus we must have Int(Fk) = ∅ for allk ∈ N implying

x ∈ X

∣∣∣∣∣∣ supj∈J

∥∥∥T j(x)∥∥∥ < ∞

=⋃k∈N

Fk

to be meager.3) We first show that (S n)n∈N is uniformly bounded, i.e., supn∈N ‖S n‖ < ∞.By assumption, the limit limn→∞ S n(x) =: S (x) exists for all x ∈ X. There-fore we have in particular limn→∞ ‖S n(x)‖Y = ‖S (x)‖Y for all x ∈ X whichin turn implies supn∈N ‖S n(x)‖Y < ∞ for all x ∈ X, i.e., (S n)n∈N is pointwisebounded and, by item 1), uniformly bounded, i.e., supn∈N ‖S n‖ < ∞.Now we show that S is a linear operator. We conclude for x, x ∈ X andλ ∈ R that

S (λx + x) = limn→∞

S n(λx + x) = limn→∞

(λS n(x) + S n(x)

)= λ lim

n→∞S n(x) + lim

n→∞S n(x)

= λS (x) + S (x).

2. UNIFORM BOUNDEDNESS AND THE BANACH-STEINHAUS THEOREM 11

Now choose a constant C ≥ supn∈N ‖S n‖. Then boundedness of S followsnow from the estimate

‖S (x)‖Y‖x‖X

= limn→∞

∥∥∥∥∥∥S n

(x‖x‖X

)∥∥∥∥∥∥Y

≤ supn∈N

∥∥∥∥∥∥S n

(x‖x‖X

)∥∥∥∥∥∥Y

≤ supn∈N‖S n‖ ≤ C.

For dual spaces, we obtain the following characterization of bounded sets.

Corollary 2.18. Let (X, ‖ ‖X) be a Banach space and (Y, ‖ ‖Y) a normedvector space. Then

U ⊆ B(X,R) bounded ⇔ supf∈U| f (x)| < ∞ ∀ x ∈ X.

Proof. ‘⇒’: Let U ⊆ B(X,R) be bounded, i.e., there exists C > 0 such that‖ f ‖ ≤ C for all f ∈ U. Therefore the inequality

| f (x)|‖x‖X

≤ supx∈X

∣∣∣∣∣∣ f(

x‖x‖X

)∣∣∣∣∣∣ = ‖ f ‖ ≤ C

is true for all f ∈ U. As a consequence, we get | f (x)| ≤ C ‖x‖X for allf ∈ U and all x ∈ X. This implies sup f∈U | f (x)| ≤ C ‖x‖X for all x ∈ X, i.e.,sup f∈U | f (x)| < ∞ for all x ∈ X.‘⇐’: Since R is Banach, so is B(X,R). The family ( f ) f∈U ∈ B(X,R) isby assumption pointwise bounded. According to item 1) in Theorem 2.17(Banach-Steinhaus), ( f ) f∈U is also uniformly bounded, i.e., sup f∈U ‖ f ‖ ≤ Cfor some C > 0, i.e., U is bounded in the operator norm.

The following facts are worthwhile mentioning.

Remark 2.19. 1) In Theorem 2.17 (Banach-Steinhaus), the assumption,that X is Banach, is necessary and cannot be dropped.

2) There are also proofs of Theorem 2.17 (Banach-Steinhaus)that do not use Theorem 2.7 (Baire), as done for example in[Chacon & Rafeiro & Vallejo].

Theorem 2.17 (Banach-Steinhaus) has lots of applications within functionalanalysis and other areas of mathematics as we will see.

Definition 2.20. Let X1, X2,Y be vector spaces. The map L : X1×X2 → Y iscalled a bilinear operator if, for all x1 ∈ X1 and for all X2 ∈ X2, the maps

X1 → Y, z1 7→ L(z1, x2),X2 → Y, z2 7→ L(x1, z2)

are linear operators.


Proposition 2.21 (Boundedness of bilinear operators). Let X1, X2 be Ba-nach spaces and L : X1 × X2 → R a bilinear operator that is continuousin each variable. Then L is continuous, i.e., given sequences αn → α andβn → β, we have L(αn, βn)→ L(α, β).

Proof. By linearity, it is enough to study continuity of L at the origin. Let(αn, βn)→ (0, 0). We have to show that L(αn, βn)→ 0. We define

Tn : X2 → R, η 7→ L(αn, η)

which satisfies (Tn)n∈N ∈ B(X2,R) and

limn→∞

Tn(η) = limn→∞

L(αn, η) = 0 ∀ η ∈ X2,

meaning, for all η ∈ X2, the pointwise limit limn→∞ Tn(η) = 0 exists. Byitem 3) of Theorem 2.17 (Banach-Steinhaus), we have supn∈N ‖Tn‖ < ∞.This implies the existence of C > 0 such that

|L(αn, η)|‖η‖X2

=|Tn(η)|‖η‖X2

≤ C ∀ η ∈ X2,∀ n ∈ N.

So we get in particular |L(αn, η)| ≤ C ‖η‖X2and, in particular for βn, the

estimate 0 ≤ |L(αn, βn)| ≤ C ‖βn‖X2which implies L(αn, βn) → 0 for n →

∞.

Another interesting application lies within the roam of numerics when ap-proximating integrals:

Given an interval [a, b] ⊂ R together with an nth subdivision

a =: t(n)0 < t(n)

1 < · · · < t(n)n−1 < t(n)

n := b

and associated weights w(n)0 , . . . ,w(n)

n , the following question arises: Whendoes

∑nk=0 w(n)

k f(t(n)k

)converge to

∫ b

af (t)dt for all f ∈ C0([a, b],R)?

For w(n)k = t(n)

k − t(n)k−1 we get the usual Riemann sums. Using Theorem 2.17

(Banach-Steinhaus), one can show more.

Proposition 2.22 (Szego). If both conditions

1) ∃ M > 0 :∑n

k=0

∣∣∣w(n)k

∣∣∣ ≤ M ∀ n ∈ N,

2) The sum∑n

k=0 w(n)k p

(t(n)k

)converges to

∫ b

ap(t)dt for n → ∞ for all poly-

nomials p,

hold true then the sum∑n

k=0 w(n)k f

(t(n)k

)converges to

∫ b

af (t)dt for n→ ∞ for

all f ∈ C0([a, b],R).

Proof. Left to the reader.

3. OPEN MAPPING, INVERSE MAPPING, AND CLOSED GRAPH THEOREM 13

3. Open mapping, inverse mapping, and closed graph theorem

Given two topological spaces X and Y , a continuous function f : X → Y ,and an open set U ⊆ Y , we know that the preimage f −1(U) is open. Thismotivates the following question. Under which assumptions on X, Y andf : X → Y is the image of an open set open?

Definition 2.23. Let X, Y be topological spaces. A map f : X → Y is saidto be open if f (U) ⊆ Y is open for all open U ⊆ X.

Let us look for examples of open maps.

Example 2.24. Let X, Y be topological spaces and f : X → Y a homeomor-phism. Then f is open.

Proof. f being a homeomorphism implies f and f −1 to be continuous. IfU ⊆ X is open, so is

(f −1

)−1(U) = f (U).

Are continuous maps open?

Example 2.25. f (x) := x2 is not open as map f : R → R, only as mapf : R→ R≥0.

Proof. The set f(] − 1, 1[

)= [0, 1[ is not open in R. But in R≥0, sets of the

form [0, b[ with b > 0 are open.

Thus we conclude that continuous maps are not necessarily open. Can non-continuous maps be open? Recall the sign map

sign : R→ R, sign(y) :=

1, y > 0,0, y = 0,−1, y < 0.

Example 2.26. The map f : R2 → R, f (x, y) := x+sign(y) is noncontinuousbut open.

Proof. The map f is clearly not continuous. Recall that open 2-dimensionalintervals of the form ]a1, b1[ × ]a2, b2[ ⊆ R2 generate the standard topologyon R2. If such an 2-dimensional interval does not intersect the line R×0 itsimage under f is of the form ]a1 +1, b1 +1[, if 0 < a2 < b2, or ]a1−1, b1−1[,if a2 < b2 < 0, and thus open. If ]a1, b1[ × ]a2, b2[ intersects R × 0 then itsimage is

]a1 + 1, b1 + 1[ ∪ ]a1, b1[ ∪ ]a1 − 1, b1 − 1[which is open.

For continuous linear operators between Banach spaces, there exists thefollowing important result.


Theorem 2.27 (Open mapping I). Let X and Y be Banach spaces and T ∈B(X,Y) surjective. Then T is open.

Before we start with the proof of Theorem 2.27 (Open mapping I), let usconsider how balls behave under linear operators.

Lemma 2.28. Let X and Y be normed vector spaces and T : X → Y a linearoperator.

1) ∀ x ∈ X, ∀ r, s > 0 :

T (B(x, r)) = T (x) + T (B(0, r)) = T (x) + rsT (B(0, s)).

2) T (B(0, r)) and T (B(0, r)) are symmetric and convex for all r > 0. Inparticular, if they contain B(y, s) for some y ∈ Y and s > 0 then theycontain also B(−y, s) and B(0, s) = 1

2 B(y, s) + 12 B(−y, s).

Proof. This follows from the linearity of T .

We now are ready for

Proof of Theorem 2.27 (Open mapping I). Step 1: Given U ⊆ X, we haveto show that T (U) ⊆ Y is open. Since open sets in normed spaces can wewritten as union of open balls it is enough to show that the image of eachopen ball is open. By Lemma 2.28, it suffices to prove that T (B(0, 1)) isopen, i.e., for all y ∈ T (B(0, 1)) there is s > 0 such that B(y, s) ⊆ T (B(0, 1)).By Lemma 2.28, it is enough to show this for y = 0.Step 2: Since T is surjective we can write Y =

⋃k∈N T (B(0, k)). As a Banach

space, Y is Baire and Theorem 2.6, item (2), implies that there must existn ∈ N such that Int

(T (B(0, n))

), ∅. Since T (B(0, n)) = nT (B(0, 1)), we

also have Int(T (B(0, 1))

), ∅. Thus there exists y ∈ Int

(T (B(0, 1))

)and r >

0 such that B(y, r) ⊆ Int(T (B(0, 1))

)⊂ T (B(0, 1)). By Lemma 2.28, we con-

clude B(0, r) ⊆ Int(T (B(0, 1))

)⊂ T (B(0, 1)) and B

(0, r

2m

)⊆ T

(B

(0, 1

2m

))for all m ∈ N.Step 3: We will show that B

(0, r

2

)⊆ T (B(0, 1)). For this purpose consider

z ∈ B(0, r

2

). Since B

(0, r

2

)⊂ T

(B

(0, 1

2

)), there is x1 ∈ B

(0, 1

2

)⊆ X such that

‖z − T (x1)‖Y <r

22 implying (z−T (x1)) ∈ B(0, r

22

)⊂ T

(B

(0, 1

22

)). Therefore

there exists x2 ∈ B(0, 1

22

)such that ‖(z − T (x1)) − T (x2)‖Y < r

23 implying

(z − T (x1) − T (x2)) ∈ B(0, r

23

)⊂ T

(B

(0, 1

23

)). By iterating this procedure,

we obtain a sequence zk :=(z −

∑ki=1 T (xi)

)∈ B

(0, r

2k+1

)⊂ T

(B

(0, 1

2k+1

))for

all k ∈ N.

3. OPEN MAPPING, INVERSE MAPPING, AND CLOSED GRAPH THEOREM 15

Since ‖zk‖Y < r2K for all k ≥ K − 1 the sequence (zk)k∈N converges with

limit 0 = limk→∞ zk = z−∑∞

i=0 T (xi) implying z =∑∞

i=0 T (xi). Moreover, theestimate ∥∥∥∥∥∥∥

k∑i=1

xi

∥∥∥∥∥∥∥X

≤

k∑i=1

‖xi‖X <

k∑i=1

12i ≤ 1

implies that yk :=∑k

i=1 xi is a Cauchy sequence. Since X is Banach, (yk)k∈N

converges and limk→∞ yk =: y satisfies ‖y‖X < 1. Since T is continuous, weconclude

T (y) = T(limk→∞

yk

)= lim

k→∞T (yk) = lim

k→∞

k∑i=1

T (xi) = z

implying B(0, r

2

)⊆ T (B(0, 1)) since z ∈ B

(0, r

2

)was chosen arbitrarily.

An immediate result is the following statement.

Theorem 2.29 (Inverse mapping). Let X and Y be Banach spaces and T ∈B(X,Y) bijective. Then T−1 ∈ B(Y, X).

Proof. By Theorem 2.27 (Open mapping I), T is open. This implies that theinverse map T−1 is continuous, i.e., T−1 is a bounded linear operator.

It is worthwhile to mention that Theorem 2.27 (Open mapping I) can bereformulated as follows.

Theorem 2.30 (Open mapping II). Let X and Y be Banach spaces andT ∈ B(X,Y) surjective. Then there exists c > 0 such that for all y ∈ Y thereis x ∈ X with ‖x‖X ≤ c ‖y‖Y and T (x) = y.

Proof. See for example [Haase].

For linear operators, being bounded is the same as being continuous. ThusTheorem 2.30 (Open mapping II) intuitively measures how much the preim-age x ∈ X of a given y ∈ Y can change if we vary y. The change is controlledby c > 0. Theorem 2.30 (Open mapping II) does not say anything about howlarge c > 0 is. Thus the existence of c > 0 is of purely theoretical value andcannot be used for explicit estimates.

Many operators are only welldefined on subspaces of the spaces one wouldlike to work on.

Definition 2.31. Let X and Y be normed vector spaces. If a linear operatorT is only defined on a subspace of X, we call this subspace the domain ofT and denote it by Dom(T ) and write

T : Dom(T ) ⊆ X → Y.


The set T (x) | x ∈ Dom(T ) is called the range or image of T and denotedby R(T ) or Im(T ).

That the domain does not coincide with the space one would like to workon, happens quite often.

Example 2.32. Consider X := f ∈ C0(R,R) | f bounded with norm‖ f ‖∞ := supx∈R | f (x)| and the differential operator T ( f ) := f ′. Then1) For Dom(T ) := f ∈ C1(R,R) | f ′ bounded, the operator T : Dom(T ) ⊂

X → X is welldefined.2) T is not bounded.

Proof. If we want that ‖ f ‖∞ and ‖T ( f )‖∞ = ‖ f ′‖∞ are finite then we haveto choose

Dom(T ) = f ∈ C1(R,R) | f ′ bounded.

T is not bounded for the following reason. Consider the sequence ( fn)n∈N

given by fn(x) := sin(nx) for which we get T ( fn)(x) = n cos(nx). Then thereis no C > 0 satisfying n = ‖T ( fn)‖∞ ≤ C ‖ fn‖∞ = C for all n ∈ N.

The operator in Example 2.32 is an often used operator, so one may ask ifthere are ‘weaker’ properties than bounded, that it still satisfies.

Definition 2.33. Let X and Y be vector spaces and T : Dom(T ) ⊂ X → Ybe a linear operator. T is closed if

graph(T ) := (x,T (x)) | x ∈ Dom(T )

is closed in X × Y.

Reformulated, this means:

Lemma 2.34. Let X and Y be vector spaces and T : Dom(T ) ⊂ X → Y bea linear operator. Then T is closed if and only if for all sequences (xn)n ∈

Dom(T ) and (yn)n∈N := (T (xn))n∈N ∈ Y with xn → x and yn → y follows thatx ∈ Dom(T ) and T (x) = y.


Using this, we find the following.

Example 2.35. The differential operator in Example 2.32 is closed.

Proof. Let ( fn)n∈N ∈ Dom(T ) = h ∈ C1(R,R) | h′ bounded and gn := f ′n =

T ( fn) ∈ h ∈ C0(R,R) | h bounded and assume fn → f and gn → g in‖ ‖∞ for some functions f and g. Since ‖ ‖∞ imposes uniform convergence,the limits f and g are continuous and bounded and we have in particularf ′ = T ( f ) = g, i.e., T is closed.

4. SEMINORMS AND FRECHET SPACES 17

Thus closed operators are not necessarily bounded. Under which assump-tions being closed and being bounded is the same, we will investigate now.

Theorem 2.36 (Closed graph). Let X, Y be Banach spaces. Then we have

T ∈ B(X,Y)⇔ T closed.

Proof. ‘⇒’: Consider Φ : X × Y → R, Φ(x, y) := ‖T (x) − y‖Y which iscontinuous. Therefore graph(T ) = Φ−1(0) is closed.‘⇐’: Consider (X × Y, ‖ ‖) with ‖(x, y)‖ := ‖x‖X + ‖y‖Y for all x ∈ X andy ∈ Y which is a Banach space. Let graph(T ) be closed. As closed subset of(X × Y, ‖ ‖), it is itself a Banach space. We consider the projections

π1 : graph(T )→ X, π1(x,T (x)) = x,π2 : graph(T )→ Y, π2(x,T (x)) = T (x).

Since

‖(x,T (x))‖ = ‖x‖X + ‖T (x)‖Y ≥ ‖x‖X = ‖π1(x,T (x))‖X ,‖(x,T (x))‖ = ‖x‖X + ‖T (x)‖Y ≥ ‖T (x)‖Y = ‖π2(x,T (x))‖Y ,

π1 and π2 are continuous. π1 is bijective such that π−11 ∈ B(graph(T ), X) by

Theorem 2.29 (Inverse mapping). Thus T = π2 π−11 is continuous.

In fact, Theorem 2.27 (Open mapping I), Theorem 2.29 (Inverse mapping),and Theorem 2.36 (Closed graph) are all equivalent. A nice application isthe following statement.

Proposition 2.37 (Hellinger-Toeplitz). Let (H, 〈 , 〉) be a Hilbert space andT : H → H a linear operator with

〈T (x), y〉 = 〈x,T (y)〉 for all x, y ∈ H.

Then T is bounded.


4. Seminorms and Frechet spaces

There are many spaces where the ‘natural norm’ is not welldefined or finite,like when passing from

(C0([0, 1],R), ‖ ‖∞

)to C0(]0, 1[,R) where the supre-

mumsnorm ‖ ‖∞ is not finite. Just picking a compact interval [a, b] ⊂ ]0, 1[and setting

p[a,b]( f ) := maxx∈[a,b]

| f (x)|

for functions f ∈ C0(]0, 1[,R) does not solve the problem since p[a,b]( f ) = 0does not imply that f ≡ 0 on the whole interval ]0, 1[. The question nowis, if exhausting ]0, 1[ by compact intervals [a, b] ⊂ ]0, 1[ may do the trickand, if yes, if this approach always works in such situations.


Definition 2.38. Let X be a vector space over K ∈ R,C. The functionp : X → R is a seminorm if1) p(x) ≥ 0 for all x ∈ X.2) p(λx) = |λ| p(x) for all x ∈ X and all λ ∈ K.3) p(x + y) ≤ p(x) + p(y) for all x, y ∈ X.

A seminorm p is no norm since p(x) = 0 does not imply x = 0. For thesame reason, setting d(x, y) := p(x− y) does not necessarily lead to a metricsince d(x, y) = 0 does not necessarily imply x = y. Nevertheless notice thatd satisfies d(x + v, y + v) = d(x, y) for all x, y, v ∈ X, i.e., d is translationinvariant. To obtain a metric, we need additional properties.

Definition 2.39. A sequence of seminorms (pk)k∈N on a vector space X isseparating if for all x ∈ X \ 0 there exists k ∈ N such that pk(x) > 0.

Now we can actually construct a metric.

Lemma 2.40. Let (pk)k∈N be a separating sequence of seminorms on thevector space X. Then

d : X × X → R≥0, d(x, y) :=∞∑

k=1

pk(x − y)2k(1 + pk(x − y))

is a translation invariant metric on X.

Proof. To prove the triangle inequality note that the function h : R≥0 → R,h(z) := z

1+z is strictly increasing. The rest are easy verifications left to thereader.

The following type of spaces is named after the French mathematician Mau-rice Frechet (1878 – 1973).

Definition 2.41. A space with a translation invariant, complete metric issaid to be a Frechet space.

To avoid confusion, let us remark the following.

Remark 2.42. The translation invariant, complete metric of a Frechet spaceis usually not induced by a norm. A Banach space can be made into aFrechet space by endowing it with the translation invariant, complete met-ric induced by its complete norm.

Before we consider examples, we need some notation.

Definition 2.43. Let U,V ⊆ Rn be open. U is compactly contained in V,written U ⊂⊂ V if the closure U is a compact subset of V.

Let us consider two standard examples.

5. THE HAHN-BANACH EXTENSION AND SEPARATION THEOREMS 19

Example 2.44. Let Ω ⊆ Rn be open.1) C0(Ω,R) can be endowed with the structure of a Frechet space.2) Lp

loc(Ω) :=f : Ω→ R

∣∣∣ f measurable,∫

U| f (x)|p dx < ∞ ∀U ⊂⊂ Ω

can

be endowed with the structure of a Frechet space for all p ∈ [1,∞[.

Proof. 1) Given Ω ⊆ Rn open, we define

(2.45) Uk :=x ∈ Ω

∣∣∣ ‖x‖eucl ≤ k, deucl(x, ∂Ω) ≥ 1k

and set pk( f ) := maxx∈Uk | f (x)| for functions f ∈ C0(Ω,R). The sequence(pk)k∈N is a separating sequence of seminorms. Using Lemma 2.40, we geta complete metric.2) We use the sets Uk from (2.45) to define for functions f ∈ Lp

loc(Ω) theseminorm

pk( f ) := ‖ f ‖Lp(Uk) :=

∫Uk

| f (x)|p dx

1p

.

The sequence (pk)k∈N is a separating sequence of seminorms. Using Lemma2.40, we get a complete metric.

5. The Hahn-Banach extension and separation theorems

This section studies the questions if it is possible to extend linear maps froma subspace to the whole vector space and, if yes, under which conditionsthis can be done, which properties the extension has and if the extensionis unique or not. Moreover, this concept can be used to find ‘separatinghyperplanes’ between two nonempty, disjoint, convex subsets. We start withthe following notation.

Definition 2.46. Let X be a vector space over the field K ∈ R,C. A linearmap f : X → K is often called a functional.

In order to extend a map, it turns out that it has to satisfy certain growthconditions which we will formulate by means of the following type of func-tions.

Definition 2.47. Let X be a vector space over R. A function p : X → R iscalled Hahn-Banach functional if1) p(x + y) ≤ p(x) + p(y) for all x, y ∈ X.2) p(tx) = tp(x) for all x ∈ X and all t ≥ 0.

Hans Hahn (1879 – 1934) was an Austrian mathematician. Note that Hahn-Banach functionals may have negative values, but behave otherwise quite


like norms. Moreover, Hahn-Banach functionals are convex since we havefor all λ ∈ ]0, 1[ and x, y ∈ X

p(λx + (1 − λ)y) ≤ p(λx) + p((1 − λ)y) = λp(x) + (1 − λ)p(y).

Example 2.48. 1) Norms and seminorms are Hahn-Banach functionals.2) Let X be a normed vector space, Ω ⊂ X open, bounded, convex with

0 ∈ Ω. Then the so-called Minkowski functional of Ω given by

pΩ : X → R≥0, pΩ(x) := infλ ≥ 0 | x ∈ λΩ

is a Hahn-Banach functional.

Hermann Minkowski (1864 – 1909) was a German mathematician. TheMinkowski functional pΩ(x) measures how much one may scale Ω in or-der to have x ‘just’ sitting in the scaled set λΩ. Now we come to the centralstatement of this section.

Theorem 2.49 (Hahn-Banach). Let X be an R-vector space and p : X → Ra Hahn-Banach functional. Let V ⊆ X be a subspace and f : V → R alinear functional satisfying f (v) ≤ p(v) for all v ∈ V. Then there exists alinear functional F : X → R satisfying

F|V = f and − p(−x) ≤ F(x) ≤ p(x) ∀ x ∈ X.

F(x) ≤ p(x) for all x ∈ X implies immediately F(−x) ≤ p(−x) and thus,by multiplying with (−1) and linearity, we get −p(−x) ≤ F(x). Therefore,in the literature, one often finds only the statement F(x) ≤ p(x). But forapplications, it is useful to consider F(x) as ‘sandwiched’ between p(x) and−p(−x).

Proof of Theorem 2.49 (Hahn-Banach). If V = X then we can choose F :=f since f (x) = − f (−x) ≥ −p(−x). Thus let us assume from now on thatV ⊂ X with V , X.Step 1: Pick x0 ∈ X \ V and consider

SpanRV, x0 = v + tx0 | v ∈ V, t ∈ R =: V0.

Then we have V ⊂ V0 with V , V0. Now we want to find some kind of‘normalized maximal distance’ between f and p. For all u, v ∈ V , we getthe estimate

f (u) + f (v) = f (u + v) ≤ p(u + v) = p(u − x0 + x0 + v)≤ p(u − x0) + p(x0 + v)

which we reformulate as

f (u) − p(u − x0) ≤ p(v + x0) − f (v) ∀ u, v ∈ V.


This yields for all u, v ∈ V

f (u) − p(u − x0) ≤ supu∈V f (u) − p(u − x0)︸︷︷︸

=:β

≤ p(v + x0) − f (v)(2.50)

Step 2: Now we want to extend f to V0. Our candidate is

f0 : V0 → R, f0(u + tx0) := f (u) + tβ

which clearly satisfies f0|V = f . We now show that f0 ≤ p in V0, i.e., wehave to show

f0(u + tx0) ≤ p(u + tx0) ∀ u ∈ V, ∀ t ∈ R

which is trivial for t = 0. Consider now t > 0, replace u and v in (2.50) byut , and multiply by t. This yields

t(

f(

ut

)− p

(ut − x0

))≤ tβ ≤ t

(p(

ut + x0

)− f

(ut

))which in turn yields

f (u) − p(u − tx0) ≤ tβ ≤ p(u + tx0) − f (u).

By definition of f0, we get

f0(u + tx0) = f (u) + tβ ≤ p(u + tx0),f0(u − tx0) = f (u) − tβ ≤ p(u − tx0)

such that f0(u + sx0) ≤ p(u + sx0) for all u ∈ U and s ∈ R, i.e., on the wholespace V0.Step 3: The idea is to repeat Step 2 until the extended subspace coincideswith X. More precisely, consider the set

U :=

(U, fU)

∣∣∣∣∣∣∣∣∣V ⊆ U ⊆ X subspace,fU : U → R linear, fU |V = f ,fU(u) ≤ p(u) ∀ u ∈ U.

carrying the partial order given by

(U, fU) (W, fW) ⇔ U ⊆ W and fW |U = fU .

By Theorem A.12 (Hausdorff maximality principle), U has a maximal el-ement

(Umax, fUmax

). We must have Umax = X since otherwise, if Umax is

strictly smaller than X, we could extend it by Step 2 in contradiction to itsmaximality. Moreover,

F := fUmax : X → R

is linear and F|V = f and F(x) ≤ p(x) for all x ∈ X by construction.

Hahn-Banach extensions are ‘compatible’ with boundedness:


Theorem 2.51 (Hahn-Banach for bounded operators). Let (X, ‖ ‖) be anormed vector space over K ∈ R,C and V ⊆ X a subspace and f ∈B(V,K). Then there is F ∈ B(X,K) with F|V = f and coinciding operatornorms ‖F‖op = ‖ f ‖op.

Proof. Case K = R: Define p(x) := ‖x‖ ‖ f ‖op which is a Hahn-Banachfunctional. Thus there is F : X → R with F|V = f and

− ‖x‖ ‖ f ‖op = −p(−x) ≤ F(x) ≤ p(x) = ‖x‖ ‖ f ‖op

which implies ‖F‖op ≤ ‖ f ‖op. Since F : X → R is an extension of f : V →R we automatically have also ‖F‖op ≥ ‖ f ‖op.Case K = C: We decompose the C-linear functional f : V → C into f =

<( f ) + i=( f ) with R-linear real and imaginary parts<( f ),=( f ) : V → R.We automatically have

∥∥∥<( f )∥∥∥

op≤ ‖ f ‖op. Using the already proven case

K = R, we get an R-linear function Φ : X → R with Φ|V = <( f ) and∥∥∥<( f )∥∥∥

op= ‖Φ‖op ≤ ‖ f ‖op. Then we define

F : X → C, F(x) := Φ(x) − iΦ(ix).

A short calculation shows<( f )(iv) = −=( f ) for all v ∈ V such that we get

F(v) = Φ(v) − iΦ(iv) = <( f )(v) − i<( f )(iv)

= <( f )(v) + i=( f )(v) = f (v)

for all v ∈ V . Now write

F(x) = |F(x)|F(x)|F(x)|

=: |F(x)| λF(x)

with λF(x) ∈ C and |λF(x)| = 1. Then we have

|F(x)| = (λF(x))−1 F(x) = F((λF(x))−1 x

)= Φ

((λF(x))−1 x

)≤ ‖Φ‖op

∥∥∥(λF(x))−1 x∥∥∥ = ‖Φ‖op ‖x‖

≤ ‖ f ‖op ‖x‖

implying ‖F‖op ≤ ‖ f ‖op. Since F : X → C is an extension of f : V → C weautomatically have ‖F‖op ≥ ‖ f ‖op, too.

Theorem 2.51 (Hahn-Banach) implies the following useful facts.

Corollary 2.52. Let K ∈ R,C and let X be a normed K-vector space. Letx, y ∈ X with x , y. Then there is F ∈ B(X,K) with F(x) , F(y).

Proof. x, y ∈ X with x , y implies z := (x − y) , 0. Consider

SpanK z = λz | λ ∈ K =: V


and f : V → K given by f (λz) := λ which implies f (x − y) = f (z) = 1. ByTheorem 2.51 (Hahn-Banach), there is F ∈ B(X,K) with F|V = f and weget

F(x) − F(y) = F(x − y) = f (x − y) = 1such that F(x) = F(y) + 1 implying F(x) , F(y).

Later on, the following statement proves to be very useful.

Corollary 2.53. Let K ∈ R,C, let (X, ‖ ‖) be a normed K-vector space,and let x ∈ X. Then there is F ∈ B(X,K) with operator norm ‖F‖op = 1and F(x) = ‖x‖.

Proof. Let x ∈ X and define V := SpanKx = λx | λ ∈ K and the K-linearmap f : V → R, f (λx) := λ ‖x‖. We get f (x) = ‖x‖ and

| f (λx)|‖λx‖

=|λ| ‖x‖|λ| ‖x‖

= 1

implying ‖ f ‖op = 1. By Theorem 2.51 (Hahn-Banach), there is F : X → Kwith F|V = f and ‖F‖op = ‖ f ‖op = 1. Moreover, we have F(x) = f (x) =

‖x‖.

This we also will need later on:

Corollary 2.54. LetK ∈ R,C and let (X, ‖ ‖) be a normedK-vector space.Moreover, let Y ⊂ X be a closed subspace with x0 ∈ X \Y. Then there existsf ∈ X∗ such that f |Y = 0, ‖ f ‖op = 1, and f (x0) = dist(x0,Y).

Proof. Cf. for instance [Alt].

Given two sets U and V with U ∩ V = ∅, a natural question to ask is ‘howmuch’ these sets are disjoint, i.e., can we find a set W with W ∩ U = ∅ =

W∩V separating U from V? If yes, what properties and shape may W have?

Since we are working in a linear setting in this chapter, it makes sense torequire of a separating set W to be ‘compatible’ in some way with linearity,meaning for example, W being a subspace of codimension 1. This require-ment in turn imposes restrictions on the possible shapes of U and V , i.e.,they also have to be ‘compatible’ with linearity in some way.

Definition 2.55. Let X be a vector space and X , 0.1) A hyperplane is a subspace V ⊂ X with V , X such that, for all sub-

spaces U with V ⊆ U ⊆ X, it follows either U = V or U = X.2) An affine hyperplane is a subset of the form V + x = v + x | v ∈ V

where V ⊂ X is a hyperplane and x ∈ X a point.

Hyperplanes in possibly infinite vector spaces are characterized similarly ashyperplanes in finite dimensions:


Remark 2.56. Let X be a vector space and X , 0.1) V ⊂ X is a hyperplane if and only if there exists a nonvanishing linear

functional f : X → R such that V = f −1(0).2) V ⊂ X is an affine hyperplane if and only if there exist a nonvanishing

linear functional f : X → R and r ∈ R such that V = f −1(r).

If we want to separate two disjoint, convex sets the following theorem de-scribes possible separating sets.

Theorem 2.57 (Separation of convex sets). Let (X, ‖ ‖) be a real normedvector space and U,V ⊂ X nonempty subsets that are disjoint and convex.1) If U is open then there exist F ∈ B(X,R) and c ∈ R such that

F(u) < c ≤ F(v) ∀ u ∈ U, ∀ v ∈ V.

2) If U is compact and V closed then there exist F ∈ B(X,R) and two realnumbers c1 < c2 such that

F(u) ≤ c1 < c2 ≤ F(v) ∀ u ∈ U, ∀ v ∈ V.

Proof. 1) Step 1: In order to apply Theorem 2.49 (Hahn-Banach), we willconstruct a Hahn-Banach functional. Pick u0 ∈ U and v0 ∈ V , set x0 :=v0 − u0, and define

Ω := U − V + x0 = (u − u0) + (v0 − v) | u ∈ U, v ∈ V.

One verifies easily that U − V is convex and therefore U − V + x0 = Ω, too.Writing Ω =

(⋃v∈V(U − v)

)+ x0, we see that Ω is open. Moreover, we have

0 ∈ Ω and x0 , Ω. The Minkowski functional pΩ : X → R≥0 of Ω is givenby

pΩ(x) = infλ ≥ 0 | x ∈ λΩ.

According to Example 2.48, it is a Hahn-Banach functional, i.e., we have inparticular for all x, y ∈ X and t ≥ 0,

pΩ(x + y) ≤ pΩ(x) + pΩ(y) and p(tx) = tp(x).

Since 0 ∈ Ω and since Ω is open, there exists r > 0 such that B(0, r) ⊆ Ω

and we get the estimate

pΩ(x) ≤‖x‖r

∀ x ∈ X.

Note that, since x0 < Ω, we have pΩ(x0) ≥ 1.Step 2: Now we construct the map that we want to extend by means ofTheorem 2.49 (Hahn-Banach). Set Y := SpanRx0 = tx0 | t ∈ R anddefine f : Y → R by f (tx0) := t. This linear functional satisfies f (x0) = 1and

f (tx0) = t ≤ tpΩ(x0) = p(tx0).


By Theorem 2.49 (Hahn-Banach), there exists a linear functional F : X →R with F|Y = f and −pΩ(−x) ≤ F(x) ≤ p(x) for all x ∈ X. The estimate

|F(x)|‖x‖

≤|p(x)|‖x‖

≤‖x‖

r ‖x‖=

1r

implies ‖F‖op ≤1r such that F is bounded.

Step 3: Given u ∈ U and v ∈ V , the points u− v + x0 lies in Ω which is openimplying pΩ(u − v + x0) < 1. Thus we get

F(u) − F(v) + 1 = F(u − v + x0) ≤ pΩ(u − v + x0) < 1

and thus F(u) < F(v) for all u ∈ U and v ∈ V implying F(U) ∩ F(V) = ∅.Since U and V are convex, so are F(U) and F(V). In particular, F(U) andF(V) are connected and form thus intervals in R. Since F is bounded and Uopen the range F(U) is by Theorem 2.27 (Open mapping I) an open interval,i.e., not containing its upper boundary point c := supu∈U F(u). Thus we haveF(u) < c ≤ F(v) for all u ∈ U and v ∈ V .2) Since U is compact, V closed and U ∩ V = ∅, the distance between Uand V

r := inf‖u − v‖ | u ∈ U, V ∈ Vis strictly positive. The collar neighbourhood

Ur := x ∈ X | ∃ u ∈ U : ‖x − u‖ < r

of U is open and convex and satisfies Ur ∩ V = ∅. Applying part 1) to Ur

and V , we obtain a bounded functional F : X → R and the existence of aconstant c2 > 0 such that

F(u) < c2 ≤ F(v) ∀ u ∈ Ur, ∀ v ∈ V.

Since U is compact and F continuous the range F(U) ⊂ R is compactcontaining its upper boundary point c1 := supu∈U F(u). Thus we obtainF(u) ≤ c1 < c2 ≤ F(v) for all u ∈ U and v ∈ V .

So far, we only addressed existence of Hahn-Banach extensions. Let us nowinvestigate under which assumptions Hahn-Banach extensions are unique.

Definition 2.58. Let (X, ‖ ‖) be a normed vector space. B(X,R) is strictlyconvex if its closed unit ball is strictly convex, i.e., for all f , g ∈ B(X,R)with operator norm ‖ f ‖op = 1 = ‖g‖op and all λ ∈ ]0, 1[, we have

‖λ f + (1 − λ)g‖op < λ ‖ f ‖op + (1 − λ) ‖g‖op = 1.

This is equivalent with requiring for B(X,R) that the closed ball centeredat the origin of radius r > 0 is strictly convex for one and, by linearity, thusany r > 0.

Theorem 2.59. Let (X, ‖ ‖) be a normed space such that B(X,R) is strictlyconvex. Then the extension in Theorem 2.51 (Hahn-Banach) is unique.


Proof. Let V ⊆ X be a subspace with V , X and let f ∈ B(V,R). Nowassume that, when applying Theorem 2.51 (Hahn-Banach), we obtain twodifferent F0, F1 ∈ B(X,R) satisfying F0|V = f = F1|V with coinciding op-erator norms ‖F0‖op = ‖ f ‖op = ‖F1‖op. W.l.o.g. we can assume by linearityand rescaling that ‖ f ‖op = ‖F0‖op = ‖F1‖op = 1. Let λ ∈ ]0, 1[ and define

Fλ := λF0 + (1 − λ)F1.

Since Fλ|V = λF0|V + (1 − λ)F1|V = λ f + (1 − λ) f = f , the functionalFλ is also an extension of f , thus we have automatically ‖Fλ‖op ≥ ‖ f ‖op.Moreover, we estimate

‖Fλ‖op = ‖λF0 + (1 − λ)F1‖op ≤ λ ‖F0‖ + (1 − λ) ‖F1‖op = 1 = ‖ f ‖op

implies ‖Fλ‖ = ‖ f ‖op = 1. This leads to a contradiction since ‖Fλ‖op < 1according to the strict convexity of B(X,R).

CHAPTER 3

Weak and weak* convergence and compactness

Banach spaces can be endowed with different types of convergences. In thissection, we define and study the three most common ones, namely the normconvergence, the weak convergence, and the weak* convergence. We aremainly interested in their compactness properties.

1. Strong, weak, and weak* convergence and reflexive spaces

If not declared otherwise, (X, ‖ ‖) is a Banach space over K ∈ R,Cthroughout this paragraph.

Definition 3.1. Given a Banach space (X, ‖ ‖), the space X∗ := B(X,K) issaid to be the dual space of X. We denote the operator norm on X∗ by ‖ ‖∗.

In the literature, the notation X′ for the dual space is also common.

Remark 3.2. Since (K, | |) is Banach, (X∗, ‖ ‖∗) is a Banach space.

There are several ‘natural’ types of convergence on X and X∗:

Definition 3.3. Let (X, ‖ ‖) be a Banach space, (X∗, ‖ ‖∗) its dual, and(xn)n∈N ∈ X and (x∗n)n∈N ∈ X∗

1) The sequence (xn)n∈N converges strongly or in norm to x ∈ X, in sym-bols xn → x, if limn→∞ ‖xn − x‖ = 0.

2) The sequence (x∗n)n∈N converges strongly or in norm to x∗ ∈ X∗, insymbols x∗n → x∗, if limn→∞

∥∥∥x∗n − x∗∥∥∥∗

= 0.3) The sequence (xn)n∈N converges weakly to x ∈ X, in symbols xn x

or xnw→ x, if limn→∞ f (xn) = f (x) for all f ∈ X∗. In this situation, x is

called the weak limiet of (xn)n∈N.4) The sequence (x∗n)n∈N converges weakly* to x∗ ∈ X∗, in symbols x∗n

∗ x∗

or x∗nw∗→ x∗, if limn→∞ x∗n(z) = x∗(z) for all z ∈ X. In this situation, x∗ is

called the weak* limiet of (x∗n)n∈N.

We observe the following relation between these types of convergence.

Proposition 3.4. 1) Strong convergence on X implies weak convergence onX.

2) Strong convergence on X∗ implies weak* convergence on X∗.

27

28 3. WEAK AND WEAK* CONVERGENCE AND COMPACTNESS

3) Weak and weak* limits are unique.

Proof. 1) Let xn → x in X. By definition, all f ∈ X∗ are continuous. Thisimplies limn→∞ f (xn) = f (x) for all f ∈ X∗, i.e., xn x.2) x∗n → x∗ in X∗ means

0 = limn→∞

∥∥∥x∗n − x∗∥∥∥∗

= limn→∞

sup ∣∣∣x∗n(z) − x∗(z)

∣∣∣ ∣∣∣ z ∈ X, ‖z‖ ≤ 1,

i.e., for all z ∈ X, we have by linearity

0 = limn→∞

∣∣∣∣∣∣x∗n(

z‖z‖

)− x∗

(z‖z‖

)∣∣∣∣∣∣ =1‖z‖

limn→∞

∣∣∣x∗n(z) − x∗(z)∣∣∣

implying x∗nw∗ x∗.

3) The weak* limit is clearly unique. Now let us assume that the weak limitis not unique, i.e., assume there are x, y ∈ X with x , y but xn x andxn y. According to Corollary 2.52, there exists f ∈ X∗ with f (x) , f (y).But the definition of weakly convergent implies f (y) = limn→∞ f (xn) = f (x)which contradicts f (x) , f (y) E.

Of course, we can iterate ‘dualizing’ spaces. The dual of the dual space isof special interest:

Definition 3.5. Let (X, ‖ ‖) be a Banach space. The space X∗∗ := (X∗)∗

endowed with its operator norm ‖ ‖∗∗ is called the bidual space of X.

The bidual space (X∗∗, ‖ ‖∗∗) is a Banach space. We are interested in it forthe following reason.

Proposition 3.6. The map

J : (X, ‖ ‖)→ (X∗∗, ‖ ‖∗∗), x 7→ J(x), J(x)( f ) := f (x) ∀ f ∈ X∗

is an embedding that is isometric, i.e., ‖J(x)‖∗∗ = ‖x‖ for all x ∈ X. Oftenthe notation x∗∗ := J(x) is used.

Proof. J is clearly linear. The estimate

‖J(x)‖∗∗ = sup |J(x)| | f ∈ X∗, ‖ f ‖∗ = 1= sup | f (x)| | f ∈ X∗, ‖ f ‖∗ = 1≤ ‖ f ‖∗ ‖x‖= ‖x‖ .

implies that J is bounded. Corollary 2.53 implies ‖J(x)‖∗∗ = ‖x‖. Moreover,J is injective: Let us assume that J(x) = J(y) for x, y ∈ X with x , y. Thismeans f (x) = f (y) for all f ∈ X∗, contradicting Corollary 2.52 E.

To the weak and weak* convergence, there are topologies associated, calledthe weak topology and the weak* topology.

2. WEAK AND WEAK* SEQUENTIALLY COMPACTNESS OF UNIT BALLS 29

Remark 3.7. 1) The weak topology is the smallest topology on X such thatall elements of X∗ are continuous. It is generated by the subbase

(x∗)−1(U) | x∗ ∈ X∗, U ⊆ K open.

2) The weak* topology is the smallest topology on X∗ such that all elementsof J(X) ⊆ (X∗)∗ are continuous on X∗. It is generated by the subbase

(J(x))−1(U) | x ∈ X, U ⊆ K open.

A natural question is if J is also surjective. The answer is ‘generally not, butsometimes well’. More precisely:

Definition 3.8. A Banach space X is reflexive if J : (X, ‖ ‖) → (X∗∗, ‖ ‖∗∗)is surjective. In the literature, one usually finds the short notation X = X∗∗

instead of J(X) = X∗∗.

Many important spaces are reflexive:

Example 3.9. 1) All finite dimensional spaces are reflexive.2) All Hilbert spaces are reflexive.3) The Lebesgue spaces Lp for 1 < p < ∞ are reflexive with (Lp)∗ = Lq for

1p + 1

q = 1.4) The Lebesgue spaces L1 and L∞ are reflexive if and only if they are finite

dimensional.

Being reflexive is nicely ‘compatible’ with many important notions:

Proposition 3.10. 1) In reflexive spaces, the weak and the weak* conver-gence of sequences in the dual space are the same.

2) Closed subspaces of reflexive spaces are reflexive.3) Let the spaces X and Y be isomorphic. Then X reflexive⇔ Y reflexive.4) A space is reflexive if and only if its dual space is reflexive.


2. Weak and weak* sequentially compactness of unit balls

We already know from previous courses that the unit ball of a metric ornormed space is compact if and only if the space is finite dimensional.

The question now is if an infinite dimensional unit ball may be compact insome sense in the weak and/or weak* topology.

Definition 3.11. 1) A set M is called weakly sequentially compact if everysequence in M admits a weakly convergent subsequence.

2) A set M is called weakly* sequentially compact if every sequence in Madmits a weakly* convergent subsequence.


The following theorem answers the question what kind of compactness wecan get without additional assumptions.

Theorem 3.12 (Banach-Alaoglu). Let (X, ‖ ‖) be a Banach space over K ∈R,C and ( fn)n∈N ∈ (X∗, ‖ ‖∗) bounded. Then ( fn)n∈N has a weakly* conver-gent subsequence whose weak* limit f ∈ X∗ satisfies ‖ f ‖∗ ≤ supn∈N ‖ fn‖∗. Inparticular, the closed unit ball in (X∗, ‖ ‖∗) is weakly* sequentially compact.

In 1932, Stefan Banach proved this statement under the assumption thatX is separable. In 1940, the Greek-Canadian-US mathematician Leonidas(Leon) Alaoglu (1914 – 1981) proved the general case.

Proof. We only proof the statement for separable spaces and refer for thegeneral case, which involves the axiom of choice, to the literature.Let (X, ‖ ‖) be a separable Banach space and S = x1, x2, x3, . . . dense inX. Let ( fn)n∈N ∈ (X∗, ‖ ‖∗) be bounded, i.e., there exists C > 0 such that‖ fn‖∗ ≤ C for all n ∈ N. We have to show that there exists f ∈ X∗ and asubsequence ( fn j) j∈N with lim j→∞ fn j(x) = f (x) for all x ∈ X.Step 1: We look for a subsequence that converges on S . Consider x1 ∈ Sand notice that | fn(x1)| ≤ C ‖x1‖ is a bounded sequence inK. By the theoremof Bolzano-Weierstraß, there exists a converging subsequence ( f j) j∈J1⊆N andwe set

limj→∞j∈J1

f j(x1) =: f (x1).

We observe that∣∣∣ fn j(x2)

∣∣∣ ≤ C ‖x2‖ is bounded for all j ∈ J1. Thus, bythe theorem of Bolzano-Weierstraß, there exists a converging subsequence( f j) j∈J2⊆J1 and we set

limj→∞j∈J2

f j(x2) =: f (x2).

Note that we still havelimj→∞j∈J2

f j(x1) = f (x1).

By iterating this procedure, we obtain a subsequence ( f j) j∈Jk that convergeson x1, . . . , xk to f (x1), . . . , f (xk). For k → ∞, we get an infinite subset I∞ ⊆N and a subsequence ( f j) j∈J∞ converging on all of S to a function f : S →K.Step 2: We now show that f : S → K is C-Lipschitz: For xr, xs ∈ S , weestimate

| f (xr) − f (xs)| = limj→∞j∈J∞

∣∣∣ f j(xr) − f j(xs)∣∣∣ ≤ sup

j∈J∞

∥∥∥ f j

∥∥∥∗‖xr − xs‖

≤ C ‖xr − xs‖ .

2. WEAK AND WEAK* SEQUENTIALLY COMPACTNESS OF UNIT BALLS 31

Since f is Lipschitz on S , we can extend f by continuity to S = X. Wedenote the extension by f : X → K.Step 3: Now we show that ( f j) j∈J∞ converges weakly* to f on X. Let ε > 0and consider an arbitrary x ∈ X. Since S is dense in X there is xk in S suchthat ‖x − xk‖ < ε. Moreover, there is N ∈ J∞ such that

∣∣∣ f j(xk) − f (xk)∣∣∣ < Cε

for all j ≥ N. Then we get for all j ∈ J∞ with j ≥ N∣∣∣ f j(x) − f (x)∣∣∣ ≤ ∣∣∣ f j(x) − f j(xk)

∣∣∣ +∣∣∣ f j(xk) − f (xk)

∣∣∣ + | f (xk) − f (x)|≤ C ‖x − xk‖ + Cε + C ‖xk − x‖< 3Cε,

meaninglimj→∞j∈J∞

f j(x) = f (x).

Since this works for all x ∈ X, the subsequence ( f j) j∈J∞ converges weakly*to f on X. Since f is the pointwise limit of a uniformly bounded familyof continuous linear operators, Theorem 2.17 (Banach-Steinhaus) impliesf ∈ X∗ with ‖ f ‖∗ ≤ sup j∈J∞

∥∥∥ j j

∥∥∥∗≤ C.

As an exercise, one can show that the closed unit ball in a Hilbert space isweakly sequentially compact. But, in fact, one can do better. Since we onlyproved Theorem 3.12 (Banach-Alaoglu) for separable Banach spaces wefeel somewhat obliged to use in the following the separable version. Thismeans that we will need the following statement, before we can study weakor weak* sequentially compactness of unit balls in reflexive spaces.

Lemma 3.13. X∗ separable⇒ X separable.

Proof. Let fn ∈ X∗ | n ∈ N be dense in X∗. For all n ∈ N, choose xn ∈ Xsuch that | fn(xn)| ≥ 1

2 ‖ fn‖∗ and ‖xn‖ = 1. Now set

Y := SpanKxn | n ∈ N ⊆ X.

Using the span over Q ⊂ R resp. over Q2 ⊂ C, we see that Y is separable.Assume that Y , X. For all f ∈ X∗ with f |Y = 0, we estimate

‖ f − fn‖∗ ≥ |( f − fn)(x)| = | fn(x)| ≥ 12 ‖ fn‖∗ ≥

12

(‖ f ‖∗ − ‖ fn − f ‖∗

)which implies 3 ‖ f − fn‖∗ ≥

12 ‖ f ‖∗. This in turn implies

3 infn∈N‖ f − fn‖∗ ≥

12 ‖ f ‖∗ .

Since ( fn)n∈N is dense in X∗ we have infn∈N ‖ f − fn‖∗ = 0 and thus ‖ f ‖∗ = 0which implies f ≡ 0 for all f ∈ X∗ with f |Y = 0. But this contradictsCorollary 2.54, so we must have Y = X, i.e., X is separable since Y isseparable.

Now we are ready to proof the following.


Theorem 3.14. Let (X, ‖ ‖) be a reflexive Banach space over K ∈ R,C.Then every bounded sequence in X has a weakly convergent subsequence.In particular, the closed unit ball of X is weakly sequentially compact.

Proof. Let (xn)n∈N ∈ X be bounded, i.e., there exists C > 0 such that ‖xn‖ ≤

C for all n ∈ N. Define

SpanKxn | n ∈ N =: Y

Since X is reflexive and Y ⊆ X a closed subspace, Y is reflexive by Propo-sition 3.10. Using the span over Q ⊂ R resp. over Q2 ⊂ C, we see thatY is separable. By Proposition 3.10, Y∗∗ is therefore reflexive, too. Since(Y∗)∗ = Y∗∗, Lemma 3.13 implies Y∗ to be separable. We use the notationJ(xn) = x∗∗n and find ∥∥∥x∗∗n

∥∥∥∗∗

= ‖ J(xn)‖∗∗ = ‖xn‖ ≤ C

such that, by Theorem 3.12 (Banach-Alaoglu), which we proved for the sep-arable case, there exists x∗∗ ∈ X∗∗ and a subsequence (x∗∗n j

) j∈N that convergesweakly* to x∗∗. We set x := J−1(x∗∗) ∈ Y and obtain

limj→∞

f (xn j) = limj→∞

x∗∗n j( f ) = x∗∗( f ) = f (x)

for all f ∈ Y∗. Moreover, since f ∈ X∗ yields f |Y ∈ Y∗ and sincex, (xn j) j∈N ∈ Y , we conclude lim j→∞ f (xn j) = f (x) for all f ∈ X∗, mean-ing (xn j) j∈N converges weakly to x.

3. Weak convergence of probability measures

We are now able to show a link between functional analysis and measuretheory by identifying weak* convergence on the dual space of continuousfunctions with weak convergence of probability measures. We start withrecalling some notation and facts from measure theory.

Let S ⊆ Rn be bounded and assume S to be in addition either open orclosed. Denote by B(S ) the σ-algebra of Borel sets which is generated bythe open subsets of S . With an additive function µ : B(S ) → Km, we canassociate the variation measure |µ| : B(S )→ [0,∞] given by

|µ| (E) := sup

k∑

i=1

|µ(Ei)|

∣∣∣∣∣∣∣∣∣k ∈ N,E1, . . . , Ek ∈ B(S ) pairwise disjoint,E1, . . . , Ek ⊆ E.

which is an additive function. The total variation of µ is defined by

‖µ‖var := |µ(S )|

and we consider the space

ca(S ,Km) := µ : B(S )→ Km | µ σ-additive, ‖µ‖var < ∞

3. WEAK CONVERGENCE OF PROBABILITY MEASURES 33

where ca stands for countably additive. We note that

(ca(S ,Km), ‖ ‖var)

is a Banach space (cf. [Alt]). A measure µ ∈ ca(S ,Km) is said to be regularif we have for all E ∈ B(S )

0 = inf |µ| (U \G) | G ⊂ E ⊂ U, G closed in S , U open in S .

We define the space

rca(S ,Km) := µ ∈ ca(S ,Km) | µ regular

and note that(rca(S ,Km), ‖ ‖var)

is a Banach space (cf. [Alt]).

Theorem 3.15 (Riesz-Radon). Let S ⊂ Rn be compact. Then

α : rca(S ,K)→(C0(S ,K)

)∗, µ 7→ α(µ)

defined by

α(µ)( f ) :=∫S

f dµ ∀ f ∈ C0(S ,K)

is a linear, continuous, isometric isomorphism.


We say that a sequence (µn)n∈N ∈ rca(S ,K) converges weakly to µ ∈rca(S ,K) if

limn→∞

∫S

f dµn =

∫S

f dµ ∀ f ∈ C0(S ,K).

A measure µ ∈ rca(S ,K) is a probability measure if µ is positive andµ(S ) = 1.

Corollary 3.16. Let S ⊂ Rn be compact. Then the subset of probabilitymeasure in rca(S ,K) is weakly sequentially compact, i.e., each sequence ofprobability measures (µn)n∈N ∈ rca(S ,K) has a subsequence (µn j) j∈N thatconverges weakly to some probability measure µ ∈ rca(S ,K).

Proof. Keep in mind that the polynomials are dense in C0(S ,K) for com-pact S ⊂ Rn, i.e., C0(S ,K) is separable. Using 3.15 (Riesz-Radon), we canidentify weak convergence of probability measures in rca(S ,K) with weak*convergence in

(C0(S ,K)

)∗. By Theorem 3.12 (Banach-Alaoglu), we getweak* sequentially compactness of the unit ball of

(C0(S ,K)

)∗. We leavethe details to the reader.

In probability theory, Corollary 3.16 is known as Prohorov’s theorem.

CHAPTER 4

Fredholm theory

Fredholm operators naturally show up in spectral theory of compact oper-ators and in applications of functional analysis to topology, geometry, andpartial differential equations. A good reference for operator theory in gen-eral and Fredholm theory in particular is [Simon, Section 3.15]. Geometricapplications are given in [McDuff & Salamon, Appendix A].

1. Definitions and examples

Let us start with the following important class of operators.

Definition 4.1. Let X,Y be Banach spaces. T ∈ B(X,Y) is compact if,for all bounded sequences (xn)n∈N ∈ X, the sequence (T (xn))n∈N ∈ Y has astrongly convergent subsequence. We denote the space of compact opera-tors from X to Y by K (X,Y).

Compact operators have many convenient properties, among others nicespectral properties concerning eigenvalues etc. For details we refer to pre-vious courses on functional analysis in the bachelor program and/or anyintroductory books on functional analysis like [Alt], [Bressan], [Haase].

The following important class of operators is named after the Swedishmathematician Erik Ivar Fredholm (1866 – 1927).

Definition 4.2. Let X,Y be Banach spaces.1) F ∈ B(X,Y) is a Fredholm operator if

• dim(ker F) < ∞.• range(F) is closed in Y.• dim(coker F) := dim

(Y/ range(F)

)< ∞.

2) The Fredholm index of a Fredholm operator F is given by

Ind(F) := dim(ker F) − dim(coker F) < ∞.

We denote the space of Fredholm operators from X to Y by F (X,Y).

When working with dimensions, one has to pay attention to the underlyingfield, i.e., if one is working over the complex or real numbers. The Fredholmindex transform via IndR = 2 IndC.

Example 4.3. The identity is a Fredholm operator.

35

36 4. FREDHOLM THEORY

In perturbation theory, operators of the form (Id−K) with K ∈ K (X,Y) areconsidered as ‘compact perturbations’ of the identity. Such operators are ofimportance in the spectral theory of compact operators since the associatedeigenvalue equation can be seen as

K − λ Id = 0 ⇔ Id− 1λK = 0 for λ ∈ C,0.

Theorem 4.4. Let X be Banach and K ∈ K (X, X). Then (Id−K) is a Fred-holm operator with Fredholm index Ind(Id−K) = 0.

Sketch of proof. Since the scope of this course does not include spectraltheory of compact operators, we only sketch the line of thought of thisquite lengthy proof and refer for details to the literature, see for instance[Bressan] (for X real Hilbert space; shorter proof) or [Alt] (for X Banachspace). Usually the claim is shown by proving the following steps succes-sively.(1) dim(ker(Id−K)) < ∞.(2) range(Id−K) is closed.(3) ker(Id−K) = 0 ⇒ range(Id−K) = X.(4) codim(range(Id−K)) ≤ dim(ker(Id−K)).(5) dim(ker(Id−K)) ≤ codim(range(Id−K)).

2. Adjoint operator on Banach spaces

Let (X, 〈 , 〉X) and (Y, 〈 , 〉Y) be Hilbert spaces and T ∈ B(X,Y). Then theHilbert adjoint operator T ∗ ∈ B(Y, X) of T is defined via the equation

〈x,T ∗(y)〉X = 〈T (x), y〉Y ∀ x ∈ X,∀ y ∈ Y.

This prompts the question wether we can define some analogue of theHilbert adjoint on Banach spaces. In particular, this requires finding somesubstitute for the scalar products that do not exist in the Banach setting.

Definition 4.5. Let X be a Banach space over K ∈ R,C and X∗ its dual.We define the dual paring of these two spaces via

〈 , 〉 : X × X∗ → K, 〈x, x∗〉 := x∗(x).

Now we are ready to extend the notion of Hilbert adjoint to Banach spaces.

Definition and Lemma 4.6. Let X, Y be Banach spaces with duals X∗,Y∗.Then

∗ : B(X,Y)→ B(Y∗, X∗), T 7→ T ∗

given by〈x,T ∗(y∗)〉 = 〈T (x), y∗〉 ∀ x ∈ X, ∀ y∗ ∈ Y∗

2. ADJOINT OPERATOR ON BANACH SPACES 37

is an isometric imbedding, i.e., coinciding operator norms ‖T‖ = ‖T ∗‖. Theoperator T ∗ is called the Banach adjoint operator of T, briefly the adjointoperator.

Proof. The map T 7→ T ∗ is surely linear. We show it being an isometry.Denote the norm on X by ‖ ‖X etc. We first show ‖T ∗‖ ≤ ‖T‖ for which weestimate

|〈x,T ∗(y∗)〉| = |〈T (x), y∗〉| ≤ ‖T‖ ‖x‖X ‖y∗‖Y∗

which implies ∣∣∣∣∣∣⟨

x‖x‖X

,T ∗(y∗)⟩∣∣∣∣∣∣ ≤ ‖T‖ ‖y∗‖Y∗

and, by taking the supremum over x ∈ X, this implies ‖T ∗(y∗)‖ ≤ ‖T‖ ‖y∗‖Y∗and therefore ‖T ∗‖ ≤ ‖T‖.To show ‖T ∗‖ ≥ ‖T‖, we estimate for all x ∈ X with ‖x‖X ≤ 1 and all y∗ ∈ Y∗

with ‖y∗‖Y∗ ≤ 1

‖T ∗‖ ≥ ‖T ∗(y∗)‖ ≥ |T ∗(y∗)(x)| = |〈x,T ∗(y∗)〉| = |〈T (x), y∗〉| .

Corollary 2.53 implies the existence of y∗ ∈ Y∗ with ‖y∗‖Y∗ = 1 and(T ∗(y∗))(x) = 〈x,T ∗(y∗)〉 = 〈T (x), y∗〉 = ‖T (x)‖Y . This yields ‖T ∗‖ ≥‖T (x)‖Y for all x ∈ X with ‖x‖X ≤ 1, hence in particular ‖T ∗‖ ≥ ‖T‖.

The Banach adjoint generalizes indeed the Hilbert adjoint as we will seenow. Recall first the following essential result from Hilbert theory namedafter the Hungarian mathematician Frigyes Riesz (1880 – 1956).

Theorem 4.7 (Riesz). Let (X, 〈 , 〉X) be a Hilbert space. Then1) ∀ y ∈ X: the map x 7→ 〈x, y〉 is a continuous linear functional.2) ∀ T ∈ B(X, X) : ∃! xT ∈ X : T (x) = 〈x, xT 〉.3) The map RX : X → X∗, x 7→ RX(x) with RX(x)(z) := 〈z, x〉X for all z ∈ X

is an isometric isomorphism.

Proof. See for example [Alt] or [Bressan].

We note

Remark 4.8. Consider the Hilbert spaces (X, 〈 , 〉X) and (Y, 〈 , 〉Y) and T ∈B(X,Y) with Hilbert adjoint T ∗ ∈ B(Y, X). Then

T ∗ = RX T∗ (RY)−1 ∈ B(Y∗, X∗).

Passing to the Banach adjoint is compatibel with taking the inverse andbeing compact:

Lemma 4.9. Let X, Y be Banach and T ∈ B(X,Y). Then1) ∃ T−1 ∈ B(Y, X) ⇔ ∃ (T ∗)−1 ∈ B(X∗,Y∗).2) T ∈ K (X,Y) ⇔ T ∗ ∈ K (Y∗, X∗).


Proof. Left as an exercise to the reader or see [Alt] or [Bressan].

Using the dual pairing we can generalize the notion of orthogonality fromHilbert spaces to Banach spaces:

Definition 4.10. Let X be a Banach space and Z ⊆ X a subspace. Then

Ann(Z) := x∗ ∈ X∗ | 〈z, x∗〉 = 0 ∀ z ∈ Z

is called the annihilator of Z.

In the literature, the annihilator of a subspace Z is sometimes also denotedby Z0 or Z⊥. It consist of all bounded linear functionals that vanish on Z.

Remark 4.11. In Hilbert spaces, the annihilator Ann(Z) of a subspace Z isisomorphic to the orthogonal complement Z⊥ of Z by Theorem 4.7 (Riesz).

Moreover, the annihilator has the following important geometric meaning.

Lemma 4.12. Let X,Y be Banach spaces and T ∈ B(X,Y). Moreover, letZ ⊆ X be a closed subspace with codim(Z) < ∞. Then

1) ker(T ∗) = Ann(range(T )).2) dim(Ann(Z) = codim(Z).

Proof. 1) We find

y∗ ∈ ker(T ∗)⇔ 0 = 〈x,T ∗(y∗)〉 = 〈T (x), y∗〉 ⇔ y∗ ∈ Ann(range(T )).

2) Let x1, . . . , xn ∈ X linear independent with X = Z ⊕ Spanx1, . . . , xn andconsider the functionals x∗1, . . . , x

∗n ∈ X∗ satisfying 〈xi, x∗j〉 = δi j (Kronecker

symbol). Then the functionals x∗1, . . . , x∗n are linearly independent and lie in

Ann(Z) implying n ≤ dim(Ann(Z)). On the other hand, given some f ∈Ann(Z) and some x = z +

∑ni=1 rixi with z ∈ Z and scalars r1, . . . , rn, we

compute

〈x, f 〉 =

n∑i=1

ri〈xi, f 〉 =

n∑i, j=1

ri〈x j, f 〉〈xi, x∗j〉 =

n∑j=1

〈x j, f 〉〈x, x∗j〉

=

⟨x,

n∑j=1

〈x j, f 〉x∗j

⟩.

Thus f can be written as linear combination of the x∗1, . . . , x∗n such that we

obtain n ≥ dim(Ann(Z)). Hence codim(Z) = n = dim(Ann(Z)).

3. FREDHOLM ALTERNATIVE 39

3. Fredholm alternative

Now we interprete equations induced by Fredholm operators under the pointof view of ‘dimension of the solution space’ and ‘dimension of solvabilityconstraints’. Roughly, the first is given by the kernel and the latter by thecokernel of the Fredholm operator. In fact, there are precisely two possiblescenarios. For that reason, one speaks of the so-called Fredholm alterna-tive:

Theorem 4.13 (Fredholm alternative). Let X be Banach and K ∈

K (X, X). Then

1) If ker(Id−K) = 0, then the homogeneous equation (Id−K)(u) = 0 hasonly the trivial solution u = 0. In this case, (Id−K) is invertible and,for each v ∈ X, there exists exactly one solution uv := (Id−K)−1(v) ∈ Xsolving the equation

(Id−K)(u) = v.

2) If ker(Id−K) , 0, then the homogeneous equation (Id−K)(u) = 0 hasnontrivial solutions. In this case, the equation

(Id−K)(u) = v

has solutions u for a given v ∈ X if and only if v satisfies the followingconstraints:

〈v, ϕ〉 = 0 for all ϕ ∈ ker(Id−K∗) = Ann(range(Id−K)).

In particular, the dimensions of the space of ‘solvability constraints’ ϕon v ∈ X coincides with the dimension of the solution space of the ho-mogeneous equation (Id−K)(u) = 0.

Proof. Let K ∈ K (X, X). By Theorem 4.4, the operator (Id−K) is Fred-holm with dim(ker(Id−K)) = dim(coker(Id−K)). Now consider:

1) If ker(Id−K) = 0 then F := (Id−K) is injective. Moreover,

0 = dim(ker(Id−K)) 4.4= dim(coker(Id−K))

implies range(Id−K) = X, i.e., (Id−K) is also surjective. By Theorem 2.29(Inverse mapping), the inverse operator exists and is continuous. This meansthat, given v ∈ X, precisely uv := (Id−K)−1(v) ∈ X solves the equation(Id−K)(u) = v.


2) We observe:

(Id−K)(u) = v has a solution u ∈ X for a given v ∈ X.⇔ v ∈ range(Id−K).⇔ 〈v, ϕ〉 = 0 for all ϕ ∈ Ann(range(Id−K)).

4.12⇔ 〈v, ϕ〉 = 0 for all ϕ ∈ ker((Id−K)∗) = ker(Id−K∗).

Moreover, since (Id−K) is Fredholm, dim(ker(Id−K)) is finite dimensionaland range(Id−K) ⊆ X is closed and has finite codimension. We conclude

dim(ker(Id−K)) 4.4= dim(coker(Id−K)) = codim(range(Id−K))4.12= dim(Ann(range(Id−K))) 4.12

= dim(ker((Id−K)∗))= dim(ker(Id−K∗)).

Furthermore, we have in particular

0 < dim(ker(Id−K)) = dim(ker(Id−K∗)) < ∞.

4. Persistence under perturbations

So far, we only had compact perturbations of the identity as explicit exam-ples for Fredholm operators. In this paragraph, we are interested in moregeneral Fredholm operators. This raises the question how to prove that anoperator is Fredholm. A major help is the following statement.

Lemma 4.14 (Semi-Fredholm estimate). Let (X, ‖ ‖X), (Y, ‖ ‖Y) and (Z, ‖ ‖Z)be Banach spaces, D ∈ B(X,Y) and K ∈ K (X,Z). Assume that there existsc > 0 such that

(4.15) ‖x‖X ≤ c(‖D(x)‖Y + ‖K(x)‖Z) ∀ x ∈ X.

Then range(D) is closed and dim(ker(D)) < ∞.

Proof. We first show dim(ker(D)) < ∞. This is equivalent to showing thatthe unit ball of ker(D) is compact. In metric spaces, this is equivalent toshowing that the unit ball of ker(D) is sequentially compact in the topologyinduced by ‖ ‖X. To show this, let (xn)n∈N ∈ ker(D) with ‖xn‖X ≤ 1. Since(xn)n∈N is bounded and K compact, the sequence (K(xn))n∈N has a convergentsubsequence (K(xn j)) j∈N which is in particular a Cauchy sequence. SinceD(xn j) = 0 for all j ∈ N, we conclude from∥∥∥xn j − xnk

∥∥∥X≤ c

(∥∥∥D(xn j) − D(xnk)∥∥∥

Y+

∥∥∥K(xn j) − K(xnk)∥∥∥

Z

)= c

∥∥∥K(xn j) − K(xnk)∥∥∥

Z

4. PERSISTENCE UNDER PERTURBATIONS 41

that (xn j) j∈N is a Cauchy sequence as well. Since X is Banach, there existsx ∈ X with lim j→∞ xn j = x. Since ker(D) is closed, we have x ∈ ker(D).

Now we show that range(D) is closed. Let (xn)n∈N ∈ X with limn→∞ D(xn) =:y ∈ Y . We have to show y ∈ range(D). The sequence (xn)n∈N is eitherbounded or unbounded in the norm topology.Let us first study the case (xn)n∈N bounded. Then (K(xn))n∈N has a convergentsubsequence (xn j) j∈N which is a Cauchy sequence due to∥∥∥xn j − xnk

∥∥∥X≤ c

(∥∥∥D(xn j) − D(xnk)∥∥∥

Y+

∥∥∥K(xn j) − K(xnk)∥∥∥

Z

).

Since X is Banach, there exists lim j→∞ xn j =: x and we get by continuityy = lim j→∞ D(xn j) = D(x).Now we show that such an (xn)n∈N cannot be unbounded. Assume that(xn)n∈N is unbounded. Then there exists a subsequence (xn j) j∈N withlim j→∞

∥∥∥xn j

∥∥∥ = ∞. Since ker(D) is finite dimensional, X splits into X =

ker(D) ⊕ X0 with some space X0. Since we are interested in range(D), wecan assume w.l.o.g. that (xn j) j∈N ∈ X0. Then u j :=

xn j∥∥∥∥xn j

∥∥∥∥X

∈ X0 has∥∥∥u j

∥∥∥X

= 1

and satisfies due to (4.15)∥∥∥u j

∥∥∥X≤ c

(∥∥∥D(u j)∥∥∥

Y+

∥∥∥K(u j)∥∥∥

Z

)= c

∥∥∥D(xn j)

∥∥∥Y∥∥∥xn j

∥∥∥X

+∥∥∥K(u j)

∥∥∥Z

.The limits lim j→∞ D(xn j) = y and lim j→∞

∥∥∥xn j

∥∥∥ = ∞ imply that

lim j→∞

∥∥∥∥D(xn j )∥∥∥∥

Y∥∥∥∥xn j

∥∥∥∥X

= 0. Since (u j) j∈N is bounded (K(u j)) j∈N has a converging

subsequence (K(u jk))k∈N and we conclude as above that (u jk)k∈N is a Cauchysequence in X0 with limk→∞ u jk =: u. It satisfies ‖u‖X = 1 and D(u) = 0.Thus we have u ∈ X0 ∩ ker(D) = 0 implying u = 0 in contradiction to‖u‖X = 1.

We want to show that being Fredholm persists under perturbation with oper-ators having sufficiently small operator norm as well as under perturbationswith arbitrary compact operators. We start with

Lemma 4.16. Let X,Y be Banach spaces, D ∈ B(X,Y) with closed range(D)and dim(ker(D)) < ∞, and K ∈ K (X,Y). Then1) range(D + K) is closed and dim(ker(D + K)) < ∞.2) There is ε > 0 such that, for all P ∈ B(X,Y) with ‖P‖ < ε, the subspace

range(D + P) is closed and dim(ker(D + P)) < ∞.3) One may replace everywhere in this lemma ker by coker and the state-

ments continue to hold true.

Proof. We want to use Lemma 4.14 (Semi-Fredholm estimate), but to doso, we need an inequality comparable to (4.15).


Let n := dim(ker(D)) and extend the identification ker(D) ' Rn by meansof Theorem 2.51 (Hahn-Banach) to an operator L ∈ B(X,Rn). Define

T : X → Y ⊕ Rn, T (x) := (D(x), L(x))

and consider the splitting X =: X⊕ker(D). The operator T is injective since,assuming T (x) = T (y), we find for x = x ⊕ xD, z = z ⊕ zD ∈ X ⊕ ker(D)

(D(x), L(xD)) = T (x) = T (z) = (D(z), L(zD))

implying xD = zD and x = z such that x = z. Moreover, since range(D) isclosed,

range(T ) = (D(x), L(xD)) | x ⊕ xD ∈ X ⊕ ker(D) ' range(D) × Rn

is closed w.r.t. the topology generated by the norm ‖ ‖ := ‖ ‖Y + ‖ ‖euclon Y ⊕ Rn. As closed subspace of the Banach space Y ⊕ Rn, range(T )is also Banach. Theorem 2.29 (Inverse mapping) yields the existence ofT−1 ∈ B

(range(T ), X

)which is in particular bounded, i.e., there is c > 0

such that∥∥∥T−1(y)

∥∥∥X≤ c ‖y‖. Written as y = T (x) = (D(x), L(xD)) we get

‖x‖X =∥∥∥T−1(T (x))

∥∥∥X≤ c ‖T (x)‖ = c

(‖D(x)‖Y + ‖L(xD)‖eucl

).

Since dim(range(L)) < ∞, the operator L is compact. Thus we have nowLemma 4.14 (Semi-Fredholm estimate) at our disposal.

Now let K ∈ K (X,Y) and P ∈ B(X,Y) and consider the operators

F1 := (D + K,K, L) : X → Y ⊕ Y ⊕ Rn,

F2 := (D + P, P, L) : X → Y ⊕ Y ⊕ Rn.

Assuming ‖P‖op small enough, one can deduce analogously as above theexistence of c1, c2 > 0 such that

‖x‖X ≤ c1(‖(D + K)(x)‖Y + ‖K(x)‖Y + ‖L(x)‖eucl

),

‖x‖X ≤ c2(‖(D + P)(x)‖Y + ‖P(x)‖Y + ‖L(x)‖eucl

).

Using ‖P(x)‖Y ≤ ‖P‖op ‖x‖X, we can transform the second equation into

‖x‖X ≤c2

1 − c2 ‖P‖op

(‖(D + P)(x)‖Y + ‖L(x)‖eucl

).

Lemma 4.14 (Semi-Fredholm estimate) implies now that D + K and D + Phave closed range and finite dimensional kernel.

We will see now that being Fredholm persists when passing to the adjointoperator. Moreover, the Fredholm index behaves additive under concatena-tion of Fredholm operators. Furthermore, Fredholm operators are intimatelyrelated to compact operators.

Proposition 4.17. Let X, Y, Z be Banach spaces and F ∈ B(X,Y).

5. INVERSE AND IMPLICITE FUNCTION THEOREMS AND SARD’S THEOREM 43

1) F ∈ F (X,Y) if and only if there exists T ∈ B(Y, X) such that (F T − Id)and (T F − Id) are compact operators.

2) F1 ∈ F (X,Y) and F2 ∈ B(Y,Z) implies F2 F1 ∈ B(X,Z) and

Ind(F2 F1) = Ind(F1) + Ind(F2).

3) F ∈ F (X,Y) ⇔ F∗ ∈ F (Y∗, X∗). In particular

Ind(F∗) = − Ind(F).

Proof. Left to the reader as an exercise.

We finally show that being Fredholm is stable under small perturbations aswell as under arbitrary compact perturbations.

Theorem 4.18. Let X,Y be Banach spaces and F ∈ F (X,Y). Then1) ∀ K ∈ K (X,Y) : (F + K) ∈ F (X,Y) and Ind(F + K) = Ind(F).2) ∃ ε > 0 : ∀ T ∈ B(X,Y) with ‖T‖ < ε :

(F + T ) ∈ F (X,Y) and Ind(F + T ) = Ind(F).

Proof. See for example [McDuff & Salamon, Appendix A].

Topologically, Theorem 4.18 means

Corollary 4.19. Let X,Y be Banach spaces. Then F (X,Y) is open inB(X,Y) w.r.t. to the topology generated by the operator norm. Moreover,the Fredholm index is constant on each connected component of F (X,Y).

5. Inverse and implicite function theorems and Sard’s theorem

In this section, we discuss the importance of Fredholm theory for differen-tial calculus between Banach spaces. We begin with

Definition 4.20. Let (X, ‖ ‖X) and (Y, ‖ ‖Y) be Banach spaces and U ⊆ Xopen. f : U → Y is (Frechet) differentiable in x ∈ U if there existsTx ∈ B(X,Y) such that

limh→0

‖ f (x + h) − f (x) − Tx(h)‖Y‖h‖X

= 0.

If such a Tx exists it is usually denoted by D f |x or d f (x) and called the(Frechet) derivative of f in x ∈ U. The map f is (Frechet) differentiableif f is (Frechet) differentiable in all x ∈ U.

Higher regularity is defined as follows.

Definition 4.21. Let X,Y be Banach spaces, U ⊆ X open and f : U → YFrechet differentiable. f is Ck for k ∈ N≥1 if the following map is Ck−1 :

U → B(X,Y), x 7→ D f |x


There is also a weaker notion of differentiability that resembles the ‘direc-tional derivative in all directions’ h ∈ X:

Definition 4.22. Let X,Y be Banach spaces over K ∈ R,C and U ⊆ Xopen. f : U → Y is Gateaux differentiable in x ∈ U if the limit

limt→0t∈K

1t ( f (x + th) − f (x))

exists for all h ∈ X.

We note the following.

Remark 4.23. Frechet differentiable ⇒: Gateaux differentiable.

The Inverse function theorem generalizes verbatim to Banach spaces:

Theorem 4.24 (Inverse function theorem). Let X,Y be Banach spaces andU ⊆ X open, and f : U → Y a Ck-map with k ∈ N≥1. Let x0 ∈ U withD f |x0 ∈ B(X,Y) bijective. Then there exists an open neighbourhood U0 ⊆

U of x0 such that the restriction f |U0 : U0 → Y is injective, V0 := f (U0) isopen in Y and(

f |U0

)−1 : V0 → U0 is Ck and (D f −1)|y =(D f | f −1(y)

)−1∀ y ∈ V0.

Proof. Cf. appendix of [McDuff & Salamon].

When generalizing the Implicite function theorem, things get more interest-ing. First we define

Definition 4.25. Let X,Y be Banach spaces, U ⊆ X open and pathcon-nected, and f : U → Y a Ck-map.1) f is a Fredholm map if D f |x is a Fredholm operator for all x ∈ U.2) If f is a Fredholm map we define its Fredholm index via

Ind( f ) := Ind(D f |x) ∀ x ∈ U

which is welldefined due to Corollary 4.19.

Moreover, we need

Definition 4.26. Let X,Y be Banach spaces, U ⊆ X open and f : U → Ya Ck-map. y ∈ Y is a regular value of f if D f |x ∈ B(X,Y) is surjective forall x ∈ f −1(y) ⊆ U.

The following theorem displays the geometric implications Fredholm mapshave.

Theorem 4.27 (Implicite function theorem). Let X,Y be Banach spaces,U ⊆ X open, f : U → Y a Ck-map with k ∈ N≥1, and y ∈ Y a regular value

5. INVERSE AND IMPLICITE FUNCTION THEOREMS AND SARD’S THEOREM 45

of f . Then M := f −1(y) ⊆ X is a Ck-Banach manifold whose tangent spacesatisfies

TxM = ker D f |x ∀ x ∈ M.If f is Fredholm then M is a finite dimensional manifold where the dimen-sion of the connected component of M containing x is given by

dim(TxM) = dim(ker(D f |x)) < ∞.


Theorem 4.27 (Implicite function) is often used to show that solution spacesof partial differential equations (PDE) are smooth manifolds. The idea is toconsider the PDE F(u) = 0 as map F between suitable Banach spaces andshow that it is in fact a Fredholm map. Then the solution space

u | F(u) = 0 = F−1(0)

is a smooth manifold of dimension Ind(F) if 0 is a regular value for F.

It remains to inquire how ‘typical’ it is for a value y ∈ Y to be regular.

Theorem 4.28 (Sard’s Theorem). Let X,Y be separable Banach spaces,U ⊆ X open, f : U → Y a Ck-map with k ≥ max1, Ind( f ) + 1. Then

Yreg( f ) := y ∈ Y | y regular value of f

is of second Baire category, i.e., it is a countable intersection of open anddense sets.


This means that ‘most’ values of Fredholm maps are regular.

CHAPTER 5

Sobolev spaces and regularity theory

A very good reference for Sobolev theory is [Adams & Fournier] or thefirst edition of that book with Adams as single author. The embedding the-orems presented in this chapter can also be found in [Alt], but the proofsthere are often quite cumbersome.

Our conventions are N := 1, 2, 3, . . . and N0 := N ∪ 0 and, throughoutthis chapter, we assume m,m1,m2, k ∈ N0 and p, p1, p2 ∈ N ∪ ∞ such thatexpressions like

1 ≤ m or 1 < p < ∞

stand form ∈ 1, 2, 3, . . . and p ∈ 2, 3, 4, . . . .

1. Definition and basic properties of Sobolev spaces

We recall some notions from measure theory. Let m ∈ N0 and p ∈ N ∪ ∞and Ω ⊆ Rn open and let λ be the Lebesgue measure on Ω. The Lebesguespace of regularity p is given by

Lp(Ω) := f : Ω→ R| f is λ-measurable, ‖ f ‖Lp(Ω) < ∞/∼

wheref ∼ g ⇔ f = g λ-almost everywhere

and, for p < ∞,

‖ f ‖Lp(Ω) :=

∫Ω

| f |p dλ

1p

and, for p = ∞,‖ f ‖L∞(Ω) := sup

x∈Ω\Nλ(N)=0

| f (x)| .

The elements of a Lebesgue space are called Lebesgue functions.

Remark 5.1. (Lp(Ω), ‖ ‖Lp(Ω)) is a Banach space.

47

48 5. SOBOLEV SPACES AND REGULARITY THEORY

A multi index of length k ∈ N0 on Rn is given by a tuple

s = (s1, . . . , sk) ∈ 0, 1, . . . , nk

and we denote by |s| := k the length of s. Given a smooth function f , weabbreviate de higher partial derivatives via

∂s f := ∂s1∂s2 . . . ∂sk f

and define for f ∈ C∞(Ω,R) the norm

‖ f ‖m,p :=∑|s|≤m

‖∂s f ‖Lp(Ω) .

The space

Xm,p := Xm,p(Ω) := f ∈ C∞(Ω,R) | ‖ f ‖m,p < ∞.

endowed with the norm ‖ ‖m,p is not complete. To obtain a Banach space,we pass to its completion

Xm,p := Xm,p(Ω) := ( f j) j∈N | ( f j) j∈N Cauchy sequence in Xm,p(Ω)/∼

where( f j) j∈N ∼ (g j) j∈N ⇔ lim

j→∞

∥∥∥ f j − g j

∥∥∥m,p

= 0.

We endow it with the metric∥∥∥( f j) j∈N − (g j) j∈N

∥∥∥Xm,p := lim

j→∞

∥∥∥ f j − g j

∥∥∥m,p

resp. norm ∥∥∥( f j) j∈N

∥∥∥Xm,p := lim

j→∞

∥∥∥ f j

∥∥∥m,p

.

Remark 5.2. 1) (Xm,p, ‖ ‖Xm,p) is complete, thus Banach.2) J : Xm,p → Xm,p, J( f ) := ( f , f , f , f , . . . ) is isometric and injective.3) Xm,p is dense in Xm,p since ( f j) j∈N ∈ Xm,p is for example approximated

by the sequence J( f1), J( f2), J( f3), . . .

The transition from Xm,p to Xm,p yields a Banach space, but its topologicaldefinition is cumbersome to work with for geometry and analysis purposes.Let us look for a more analysis/geometry friendly description.

Before we give the actual definition, let us gain some intuition. ( f j) j∈N beingan element of Xm,p means that ( f j) j∈N is a Cauchy sequence in Xm,p, i.e.,

∀ ε > 0 ∃ N ∈ N : ∀ j, k ≥ N :∥∥∥ f j − fk

∥∥∥m,p

< ε

Because of∥∥∥ f j − fk

∥∥∥m,p

=∑|s|≤m

∥∥∥∂s f j − ∂s fk

∥∥∥Lp(Ω)

, the sequences (∂s f j) j∈N

are Cauchy sequences in the Banach spaces (Lp(Ω), ‖ ‖Lp(Ω)) for all |s| ≤ m.

1. DEFINITION AND BASIC PROPERTIES OF SOBOLEV SPACES 49

Thus lim j→∞ ∂s f j =: f (s) ∈ Lp(Ω) exists w.r.t. convergence in ‖ ‖Lp(Ω) and inparticular lim j→∞ f j =: f (0) =: f . For all |s| ≤ m and for all

ϕ ∈ C∞c (Ω,R) := ϕ ∈ C∞(Ω,R) | ϕ has compact support,

we obtain by partial integration∫Ω

f j ∂sϕ dλ = (−1)|s|∫Ω

(∂s f j) ϕ dλ

Using the theorem of dominated convergence, we can pass to the Lp-limitsand obtain the identity∫

Ω

f ∂sϕ dλ = (−1)|s|∫Ω

f (s)ϕ dλ.(5.3)

If f is smooth then f (s) coincides with the derivative ∂s f . If f is not dif-ferentiable in the classical sense we set ∂s f := f (s) and speak of a weakderivative. This motivates the definition of the space

Hm,p(Ω) :=

f ∈ Lp(Ω)

∣∣∣∣∣∣∣∣∣∣∀ |s| ≤ m : ∃ f (s) ∈ Lp(Ω) : f (0) = f and∫Ω

f ∂sϕ dλ = (−1)|1|∫Ω

f (s)ϕ dλ ∀ ϕ ∈ Cc(Ω,R)

which is often also denoted by Wm,p and called Sobolev space of regularity(m, p). Its elements are called Sobolev functions of regularity (m, p). Wedefine

H0,p(Ω) := Lp(Ω).We endow Hm,p(Ω) with the norm

‖ f ‖Hm,p(Ω) :=∑|s|≤m

∥∥∥ f (s)∥∥∥

Lp(Ω).

and we callm −

np

the Sobolev number of the Sobolev space Hm,p(Ω). It is motivated by theintegrability properties of the function x 7→ ‖x‖ρ with ρ ∈ R around theorigin, cf. for example [Bressan, Example 8.23] or [Alt, Section 10.7].

Lemma 5.4. C∞(Ω)∩Hm,p(Ω) is dense in Hm,p(Ω) for 1 ≤ p < ∞ and 0 ≤ m.For p = ∞, this is not true.

Proof. This is proven using convolution with smooth functions with com-pact support. The details are left to the reader or see [Alt, Theorem3.28].

Now we come to the relation between Xm,p(Ω) and Hm,p(Ω).


Proposition 5.5. For all 1 ≤ p ≤ ∞ and all 1 ≤ m,

J : Xm,p(Ω)→ Hm,p(Ω), J(( f j) j∈N) := limj→∞

f j (limit taken in Lp(Ω))

is isometric and injective. If 1 ≤ p < ∞ and 1 ≤ m, then J is surjective.

Proof. J is injective since, given ( f j) j∈N, (g j) j∈N ∈ Xm,p with J(( f j) j∈N) =

J((g j) j∈N), we get

0 = J(( f j − g j) j∈N) = limj→∞

( f j − g j) = f − g as Lp(Ω)-limiet.

J is isometric: We consider∥∥∥( f j) j∈N

∥∥∥Xm,p(Ω)

= limj→∞

∥∥∥ f j

∥∥∥Xm.p(Ω)

= limj→∞

∑|s|≤m

∥∥∥∂s f j

∥∥∥Lp(Ω)

=∑|s|≤m

limj→∞

∥∥∥∂s f j

∥∥∥Lp(Ω)

Being a Cauchy sequence in Lp(Ω), the sequence (∂s f j) j∈N converges inLp(Ω). Since the identity (5.3) determines the functions f (s) uniquely inLp(Ω), we conclude

· · · =∑|s|≤m

∥∥∥ f (s)∥∥∥

Lp(Ω).

For p , ∞, we conclude surjectivity of J via Lemma 5.4.

Sobolev spaces are very convenient to work with for the following reason.

Proposition 5.6. Let Ω ⊆ Rn be open and 0 ≤ m and 1 ≤ p ≤ ∞. Then(Hm,p(Ω), ‖ ‖Hm,p(Ω)) is a Banach space.

Proof. If ( f j) j∈N is a Cauchy sequence in Hm,p(Ω) so are the weak deriva-tives (∂s f j) j∈N in Lp(Ω). Since Lp(Ω) is Banach, the weak derivatives con-verge to some f (s) ∈ Lp(Ω). In particular, ( f j) j∈N converges to somef := f (0) ∈ Lp(Ω). By definition, f ∈ Hm,p(Ω) and the f (s) are the weakderivatives of f .

If p , ∞, Proposition 5.6 can also be deduced from Proposition 5.5. We canalso define Sobolev spaces with boundary value zero: For Ω ⊆ Rn open,0 ≤ m and 1 ≤ p < ∞ (note p , ∞!) we define

Hm,p (Ω) := f ∈ Hm,p(Ω) | ∃ ( f j) j∈N ∈ C∞c (Ω,R) : limj→∞‖ f − fk‖Hm,p(Ω) = 0

and endow it with the norm ‖ ‖Hm,p(Ω). Then

Hm,p (Ω) is a closed subspaceof Hm,p(Ω) and therefore a Banach space. It is compatible with extension of

2. WEAK AND STRONG CONVERGENCE IN SOBOLEV SPACES 51

functions in the following sense. Let Ω, Ω ⊆ Rn open with Ω ⊆ Ω. Given a

function f ∈

Hm,p (Ω), the function

f := f on Ω,

0 on Ω \Ω

can be approximated by functions from C∞c (Ω,R) and is thus an element of

Hm,p (Ω).

2. Weak and strong convergence in Sobolev spaces

Strong convergence always implies weak convergence as we saw in Propo-sition 3.4, but sometimes one can pass from weak back to strong conver-gence at a certain ‘cost’. This is possible in Sobolev spaces and the ‘priceto pay’ is the loss of a derivative, i.e., loss of regularity, as we will see now.

Regularity of Sololev functions depends to a certain extend on controllingthe functions near and on the boundary of the underlying open set Ω ⊆ Rn.On the one hand, the requirements on the boundary should be sufficientlystrong to allow reasonable proofs, but, on the other hand, loose enough toadmit the domains of standard PDE problems which are often of rectangularshape. Thus requiring Lipschitz regularity for the boundary is a reasonablechoice.

Definition 5.7. Let Ω ⊆ Rn be open and bounded. Ω is said to have Lip-schitz boundary if its boundary ∂Ω can be covered by a finite number ofopen sets U1, . . . ,Uk such that we have for all 1 ≤ j ≤ k:

• Ω lies on precisely one side of ∂Ω ∩ U j in Ω ∩ U j.• ∂Ω ∩ U j can (maybe after a rotation) be seen as the graph of a

Lipschitz function.

We note

Remark 5.8. Weak convergence of a sequence (uk)k∈N in Hm,p(Ω) is equiv-alent to weak convergence of the (weak) derivatives ∂suk in Lp(Ω) for all|s| ≤ m.

The following result is named after the Austrian-German mathematicianFranz Rellich (1906 – 1955).

Theorem 5.9 (Rellich). Let Ω ⊆ Rn be open and bounded with Lipschitzboundary. Let 1 ≤ p < ∞ and 1 ≤ m and (uk)k∈N ∈ Hm,p(Ω) and u ∈Hm−1,p(Ω). Then1) If (uk)k∈N ∈ Hm,p(Ω) is bounded and uk u converges weakly in Hm,p(Ω)

then uk → u converges strongly in Hm−1,p(Ω).


2) Statement 1) also holds true with Hm,p(Ω) and Hm−1,p(Ω) replaced by

Hm,p (Ω) and

Hm−1,p (Ω). The hypothesis ‘Lipschitz boundary’ is in thissituation not necessary.

Proof. See for example [Alt, Sections A 8.1 and A 8.4].

Thus the price we pay for gaining strong convergence from weak conver-gence is the loss of one (weak) derivative.

Values of Sobolev functions are welldefined up to change on a set ofLebesgue measure zero in the domain of definition. If we want to studySobolev functions on sets of measure zero – like Lipschitz boundarys – wehave to make sure that the function is welldefined on the zero set as ele-ment of some function space. If the equivalence class contains continuousor even smooth representatives there is no problem, but this is not always thecase. Geometrically, the following statement describes when ‘weak bound-ary values’ are welldefined.

Theorem 5.10 (Trace operator). Let Ω ⊆ Rn be open and bounded withLipschitz boundary and let 1 ≤ p ≤ ∞. Then there exists exactly oneoperator T : H1,p(Ω) → Lp(∂Ω) such that T (u) = u|∂Ω for all u ∈C0(Ω,R) ∩ H1,p(Ω). This operator is called Trace operator.

Proof. See for example [Alt, Section A 8.6].

Thus the price we pay for gaining (weak) boundary values is the loss of one(weak) derivative. The name Trace operator has noting to do with the traceof a matrix. It rather describes the ‘trace’ of ∂Ω as ‘curve’ in the graph of aSobolev function.

Theorem 5.11 (Rellich’s trace theorem). Let Ω ⊆ Rn be open and boundedwith Lipschitz boundary. Let 1 ≤ p < ∞ and u, (uk)k∈N ∈ Hm,p(Ω). If uk uconverges weakly in Hm,p(Ω) then uk → u converges strongly in Lp(∂Ω).

Proof. See for example [Alt, Section A 8.13].

3. Holder spaces

We want to investigate the interaction between Sobolev and Lebesgue func-tions on the one hand and ‘classical’ properties like continuity, differentia-bility, and smoothness on the other hand. Before we delve into this quest,we first generalize the notion of ‘Lipschitz’.

Let (Y, | |) be a Banach space, Ω ⊆ Rn and ‖ ‖ a norm on Rn. Let α ∈ ]0, 1]and f : Ω → Y a function. We define the Holder constant of f on Ω with

4. REGULARITY AND COMPACTNESS VIA EMBEDDING THEOREMS 53

Holder coefficient α as

holα( f ,Ω) := sup| f (x) − f (y)|‖x − y‖α

∣∣∣∣∣ x, y ∈ Ω, x , y∈ [0,∞].

For α = 1, the Holder constant is the Lipschitz constant. For open andbounded Ω ⊆ Rn and k ∈ N0 and α ∈ ]0, 1], the Holder spaces are definedas

Ck,α(Ω,Y):= f ∈ Ck(Ω,Y)

| holα(∂s f ,Ω) < ∞ for |s| = k.

We endow Ck,α(Ω,Y)with the norm

‖ f ‖Ck,α

(Ω,Y

) :=∑|s|≤k

‖∂s f ‖sup +∑|s|=k

holα(∂s f ,Ω).

The elements of a Holder space are called Holder functions. It makes senseto extend the definition of Holder spaces to α = 0 by setting

Ck,0(Ω,Y):= Ck(Ω,Y)

.

The Holder number of the Holder space Ck,α(Ω,R) is given by

k + α.

Remark 5.12.(Ck,α(Ω,Y)

, ‖ ‖Ck,α

(Ω,Y

)) is a Banach space.

4. Regularity and compactness via embedding theorems

On the one hand, Sobolev spaces are necessary since solutions of even‘smooth’ optimalization problems are not necesarily differentiable or con-tinuous. On the other hand, it is certainly nicer to have a ‘classical’ (= con-tinuous/ Lipschitz/ differentiable/ smooth) solution than a ‘weak’ solutionin the Sobolev sense.

Since Sobolev functions are in essence equivalence classes of functions wewould like to know if/when such an equivalence class contains ‘classical’representatives. In the end, it comes down to comparing the ‘total regular-ity’ of the spaces which is given by their Sololev numbers and/or Holdernumbers. To this end, we now present a number of theorems from the liter-ature.

Theorem 5.13 (Ck+1 → Holder). Let Ω ⊆ Rn be open and bounded withLipschitz boundary. Then, for all k ∈ N0,

Id : Ck+1(Ω,R)→ Ck,1(Ω,R)

is welldefined and continuous.

Proof. See for example [Alt, Section 10.5].

Moreover


Theorem 5.14 (Holder → Sobolev). Let Ω ⊆ Rn be open and bounded withLipschitz boundary. Then, for all k ∈ N0,

Id : Ck,1(Ω,R)→ Hk+1,∞(Ω,R)

is welldefined. Moreover, it is an isomorphism in the following sense: Forall u ∈ Hk+1,∞(Ω,R) there exists exactly one u ∈ Ck,1(Ω,R) such that u = uLebesgue almost everywhere, i.e., u = u in Hk+1,∞(Ω,R).


Moreover

Theorem 5.15 (Holder → Holder). Let Ω ⊆ Rn be open and bounded. Letk1, k2 ∈ N0 and α1, α2 ∈ [0, 1] with Holder numbers satisfying

k1 + α1 > k2 + α2.

If k1 > 0, assume in addition Ω to have Lipschitz boundary. Then the em-bedding

Id : Ck1,α1(Ω,R)→ Ck2,α2(Ω,R)is a compact operator.


Moreover

Theorem 5.16 (Sobolev → Sobolev). Let Ω ⊆ Rn be open and boundedwith Lipschitz boundary. Let 0 ≤ m1,m2 and 1 ≤ p < ∞. Then1) If

m1 −np1≥ m2 −

np2

and m1 ≥ m2

then the embedding Id : Hm1,p1(Ω) → Hm2,p2(Ω) is welldefined and con-tinuous. In particular, there exists C = C(n,Ω,m1,m2, p1, p2) > 0 suchthat

‖u‖Hm2 ,p2 (Ω) ≤ C ‖u‖Hm1 ,p1 (Ω) .

2) Ifm1 −

np1

> m2 −np2

and m1 > m2

then the embedding Id : Hm1,p1(Ω)→ Hm2,p2(Ω) is compact.3) Statements analogous to items 1) and 2) hold also true for the spaces

Hm1,p1 (Ω) and

Hm2,p2 (Ω). Here the assumption ‘Lipschitz boundary’ isnot necessary.


Moreover

4. REGULARITY AND COMPACTNESS VIA EMBEDDING THEOREMS 55

Theorem 5.17 (Sobolev →Holder). Let Ω ⊆ Rn be open and bounded withLipschitz boundary. Let 1 ≤ m and 1 ≤ p < ∞ and 0 ≤ k and α ∈ [0, 1].Then1) If

m −np

= k + α and 0 < α < 1

then the embedding operator Id : Hm,p(Ω)→ Ck,α(Ω) is welldefined andcontinuous, i.e., for all u ∈ Hm,p(Ω) there exists exactly one u = Id(u) ∈Ck,α(Ω) such that u = u Lebesgue almost everywhere. In particular, thereexists C = C(n,Ω,m, p, k, α) > 0 such that

‖Id(u)‖Ck,α(Ω) ≤ C ‖u‖Hm,p(Ω) .

2) Ifm −

np> k + α

then the embedding operator Id : Hm,p(Ω)→ Ck,α(Ω) is compact.3) Statements analogous to items 1) and 2) hold also true for the space

Hm,p (Ω). Here the assumption ‘Lipschitz boundary’ is not necessary.


We summarize the geometric meaning of these embedding theorems:

Remark 5.18. 1) If we can embed ‘space one’ into ‘space two’ then eachelement of ‘space one’ can be seen as element of ‘space two’. If theelements are equivalence classes of functions, this simply means thateach equivalence class contains a representatives that lives in fact in thetarget space of the embedding.

2) As continuous operator, the embedding implies an estimate of the in-volved norms.

3) Compact embeddings are useful since composing bounded operatorswith compact embeddings adjusts the domain or target of the boundedoperator suitably to ‘turn the bounded operator compact’.

CHAPTER 6

Elliptic partial differential equations

There are three big classes of partial differential equations (PDEs), namelyelliptic, parabolic, and hyperbolic PDEs. In this chapter, we will study ellip-tic boundary value problems. More precisely, we will state the ‘classical’ (=differentiable or smooth) formulation and then turn it into a ‘weak’ versionon Sobolev spaces. Subsequently we will describe how to solve ‘weak’ el-liptic PDEs with Dirichlet boundary conditions. A typical example for thistype of problems is

(6.1)−∆u − u = f in Ω ⊆ Rn,

u = 0 on ∂Ω.

where ∆ :=∑n

k=1 ∂2xk

is the Laplace operator, the archetype of elliptic 2ndorder differential operators.

1. Classical elliptic boundary value problem

We generalize the boundary value problem in (6.1) as follows. Let Ω ⊆ Rn

be open and bounded, ai j, bi ∈ C1(Ω,R) for 1 ≤ i, j ≤ n and c, f ∈ C0(Ω,R).We define the differential operator

L : C2(Ω,R)→ C0(Ω,R),

L(u) := −n∑

i, j=1

∂x j(ai j(x)∂xiu) +

n∑i=1

∂xi(bi(x)u) + c(x)u(6.2)

and want to find a solution u ∈ C2(Ω,R) ∩ C0(Ω,R) of the boundary valueproblem L(u) = f in Ω,

u = 0 on ∂Ω.(6.3)

Boundary conditions of the form u = g on ∂Ω with g ∈ C0(∂Ω,R) arecalled Dirichlet boundary conditions and, in case of g ≡ 0, homogeneousDirichlet boundary conditions. Note that there is a transformation turninga Dirichlet boundary problem with general g into a homogeneous one, see[Bressan, Remark 9.5].

57

58 6. ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

Definition 6.4. L from (6.2) is elliptic if there exists θ > 0 such that

ξT a(x) ξ :=n∑

i, j=1

ai, j(x)ξiξ j ≥ θ ‖ξ‖2

for all x ∈ Ω and for all ξ ∈ Rn, i.e., de matrix a := (ai j)1≤i, j≤n is strictlypositive definite.

In particular, if the matrix a of an elliptic operator is diagonalizable then theabsolute values of the eigenvalues are bounded from below by θ > 0. Thename ‘elliptic’ comes from the fact that ξ ∈ Rn |

∑ni, j=1 ai, j(x)ξiξ j = const

describes under the assumption ‘elliptic’ an ellipse.

Example 6.5. Setting (ai j)1≤i, j≤n to be the unit matrix, bi ≡ 0 and c(x) ≡ 1in (6.3), we obtain the boundary value problem (6.1). We may choose θ = 1and obtain that the operator ∆u − u in (6.1) is elliptic.

Solution methods for the classical elliptic Dirichlet boundary value problemgiven by (6.3) can be found for instance in [Farlow].

Apart from being interesting on its own account, the Laplace operator ismoreover related to several other important operators:

Remark 6.6. 1) Identify R2 ' C via (x, y) = x + iy and define the operators∂ := ∂x − i∂y and ∂ := ∂x + i∂y. Then

∂ ∂ = ∂ ∂ = ∆.

∂ and ∂ are so-called Cauchy-Riemann operators.2) OperatorsD with the property

D2 := D D = ∆

are called Dirac operators. They appear for example in Clifford analy-sis where they are of the form D =

∑nj=1 e j ∂x j with orthonormal frame

e1, . . . , en of Rn.

These relations between the various operators are of importance since theoperators ∂, ∂,D ‘inherit’ certain properties from ∆ such that their analysisprofits from the analysis of ∆ and vice versa.

2. Weak elliptic boundary value problem

In this paragraph, we investigate how to solve (6.3) in case the functionsai j, bi, c, f are not any more continuously differentiable or continuous. Tothis end, let us reformulate the equation L(u) = f using multiplication by

2. WEAK ELLIPTIC BOUNDARY VALUE PROBLEM 59

test functions and integration w.r.t. the Lebesgue mesure λ on Rn. In thefollowing, we use the short notation∫

Ω

g :=∫Ω

g(x) dλ(x).

For test functions ϕ ∈ Cc(Ω,R), we obtain∫Ω

fϕ =

∫Ω

L(u) ϕ

=

∫Ω

− n∑i, j=1


n∑i=1

∂xi(bi(x)u) + c(x)u

ϕand by means of partial integration

(6.7) =

∫Ω

n∑i, j=1

ai j(x)∂xiu∂x jϕ −

n∑i=1

bi(x)u∂xiϕ + c(x)uϕ

.To assure welldefinedness of the integrals and compliance with the bound-ary condition u = 0 on ∂Ω, the following assumptions are necessary and/oruseful:

• ai j, bi, c ∈ L∞(Ω) for 1 ≤ i, j ≤ n. This gives us the freedom toadjust the conditions on u and ϕ to the geometry of the problem.• u needs at least one weak derivative. Since u and its weak de-

rivative need to be integrable while being multiplied with ϕ, ask-ing L2-integrability of u and its derivative makes sense. This sug-gests to ask for H1,2-regularity of u. In order to comply with theboundary condition u = 0 on ∂Ω, a natural choice is to work with

u ∈

H1,2(Ω).• ϕ starts out from being a smooth test function with compact sup-

port, but we may relax this assumption: ϕ needs at least one weakderivative and both ϕ and its derivative need to be integrablewhile being multiplied with another function. This suggests ask-ing for L2-integrability for ϕ and its derivative, i.e., H1,2-regularityof ϕ. Since the original ϕ had compact support, the choice of

ϕ ∈

H1,2(Ω) is suggestive. In addition, this makes u and ϕ beingelements of the same space which simplifies the analysis consider-

ably, in particular, since

H1,2(Ω) is a Hilbert space.


• f needs to be integrable while being multiplied with another func-tion. This suggests f ∈ L2(Ω). This is compatible with the regu-

larity of ϕ since the (even compact) embedding

H1,2(Ω)→ L2(Ω)assures welldefinedness of the identification

∫Ω

fϕ = 〈 f , ϕ〉L2 .

These thoughts lead to

Definition 6.8. The weak elliptic Dirichlet boundary value problem isdefined as follows.

Given: Ω ⊆ Rn open and bounded, f ∈ L2(Ω), ai j, bi, c ∈ L∞(Ω)with a := (ai j)1≤i, j≤n elliptic, i.e., there exists θ > 0 such that forLebesgue-almost all x ∈ Ω:

ξT a(x) ξ :=n∑

i, j=1

ai, j(x)ξiξ j ≥ θ ‖ξ‖2 .

Wanted: u ∈

H1,2(Ω) satisfying

(6.9)∫Ω

n∑i, j=1

ai j(x)∂xiu ∂x jϕ −

n∑i=1


=

∫Ω

fϕ

for all ϕ ∈

H1,2(Ω).

Such a Sobolev function u ∈

H1,2(Ω) is called a weak solution of the ellipticDirichlet boundary problem

L(u) = f in Ω,

u = 0 on ∂Ω

associated to the ‘formal operator’

L(u) := −n∑

i, j=1


n∑i=1

∂xi(bi(x)u) + c(x)u.

3. POSITIVE DEFINITE OPERATORS AND THE THEOREM OF LAX-MILGRAM 61

Now we want to make use of the Hilbert space structure of

H1,2(Ω) andL2(Ω). Recall the definition of the scalar products

〈g, ϕ〉L2 =

∫Ω

gϕ ∀ g, ϕ ∈ L2(Ω),

〈g, ϕ〉H1,2 = 〈g, ϕ〉L2 +

n∑i=1

〈∂xig, ∂xiϕ〉L2 ∀ g, ϕ ∈ L2(Ω)

=

∫Ω

gϕ +

n∑i=1

∫Ω

∂xig ∂xiϕ

and identify the left hand side of equation (6.9) with the bilinear form

B :

H1,2(Ω) ×

H1,2(Ω)→ R,

B(u, ϕ) :=∫Ω

n∑i, j=1

ai j(x)∂xiu ∂x jϕ −

n∑i=1


.We conclude

Corollary 6.10. u ∈

H1,2(Ω) is a weak solution of (6.9) if and only if u ∈

H1,2(Ω) solves the equation

B(u, ϕ) = 〈 f , ϕ〉L2 ∀ ϕ ∈

H1,2(Ω) .

This transforms the weak elliptic Dirichlet boundary problem into the prob-

lem of solving the equation B(u, ϕ) = 〈 f , ϕ〉L2 for u ∈

H1,2(Ω), i.e., we arenow looking for some inverse operator between Hilbert spaces.

3. Positive definite operators and the theorem of Lax-Milgram

This section provides us with tools for solving equations involving scalarproducts and bilinear forms in Hilbert spaces.

Definition 6.11. Let (H, 〈 , 〉 be a real Hilbert space with induced norm‖ ‖ :=

√〈 , 〉 and T : H → H a linear operator. T is strictly positive

definite with constant θ > 0 if

θ ‖h‖2 ≤ 〈T (h), h〉 ∀ h ∈ H.

Strictly positive operators are invertible:

Theorem 6.12 (Inverse of strictly positive definite operators). Let(H, 〈 , 〉) be a real Hilbert space. Let T ∈ B(H,H) be strictly positive


definite with constant θ > 0. Then T has an inverse T−1 ∈ B(H,H) with op-erator norm

∥∥∥T−1∥∥∥

op≤ 1

θ. In particular, for all f ∈ H, there exists a unique

‘solution’ u ∈ H such that T (u) = f , namely u = T−1( f ).

Proof. Step 1: We show that T is injective. Let ‖ ‖ :=√〈 , 〉 and estimate

θ ‖h‖2 ≤ 〈T (h), h〉 ≤ ‖T (h)‖ ‖h‖ .

This yields θ ‖h‖ ≤ ‖T (h)‖ which means that T (h) = 0 implies h = 0, i.e., Tis injective.Step 2: We show that T is surjective. To this end, we first show that range(T )is closed. Let (hk)k∈N be a sequence in H with existing limit limk→∞ T (hk).In particular, (T (hk))k∈N is a Cauchy sequence which, together with the es-timate ∥∥∥hk − h j

∥∥∥ ≤ 1θ

∥∥∥T (hk) − T (h j)∥∥∥ ,

implies that (hk)k∈N is a Cauchy sequence, too. Since H is Hilbert, the limith := limk→∞ hk exists. The continuity of T implies now limk→∞ T (hk) =

T (h) ∈ range(T ). Since range(T ) is closed, we have the orthogonal decom-position H = range(T ) ⊕ (range(T ))⊥. For all v ∈ (range(T ))⊥, the estimateθ ‖v‖2 ≤ 〈T (v), v〉 = 0 implies v = 0, hence H = range(T ).Stap 3: We just proved that T ∈ B(H,H) is bijective. Theorem 2.29 (Inversemapping) implies the existence of the inverse T−1 ∈ B(H,H). Therefore,given f ∈ H, the equation T (u) = f has the unique solution u = T−1( f ).Moreover,

θ∥∥∥T−1( f )

∥∥∥ ≤ ∥∥∥T (T−1( f ))∥∥∥ = ‖ f ‖

implies∥∥∥T−1

∥∥∥op≤ 1

θ.

Recall that a bilinear form is a mapping B : H × H → R that is linear inboth variables.

Definition 6.13. A bilinear form B : H × H → R is continuous if thereexists C > 0 such that |B(u, v)| ≤ C ‖u‖ ‖v‖ for all u, v ∈ H. A bilinear formB is strictly positive definite with constant θ > 0 if there exists θ > 0 suchthat θ ‖u‖2 ≤ B(u, u) for all u ∈ H.

The following important result is named after the Hungarian-Americanmathematician Peter D. Lax (born in 1926) and the American mathemati-cian Arthur N. Milgram (1912 – 1961).

Theorem 6.14 (Lax-Milgram). Let (H, 〈 , 〉) be a real Hilbert space andB : H × H → R a continuous bilinear form that is strictly positive definitewith constant θ > 0. Then, for all f ∈ H, there exists a unique u f ∈ H suchthat B(u f , ϕ) = 〈 f , ϕ〉 for all ϕ ∈ H. Moreover,

∥∥∥u f

∥∥∥ ≤ 1θ‖ f ‖.

4. SOLVING THE WEAK ELLIPTIC DIRICHLET BOUNDARY PROBLEM 63

Proof. For all u ∈ H, the map B(u, ·) : H → H, ϕ 7→ B(u, ϕ) is a continuouslinear functional. By Theorem 4.7 (Riesz), there exists a unique uB ∈ Hsuch that B(u, ϕ) = 〈uB, ϕ〉 for all ϕ ∈ H.Step 1: We show that

(6.15) βB : H → H, βB(u) := uB

is a strictly positive definite continuous linear operator: For r ∈ R, the iden-tity 〈rβB(u), ϕ〉 = rB(u, ϕ) = B(ru, ϕ) implies rβB(u) = βB(ru). Moreover,we find for u1, u2, ϕ ∈ H:

〈βB(u1) + βB(u2), ϕ〉 = 〈βB(u1), ϕ〉 + 〈βB(u2), ϕ〉 = B(u1, ϕ) + B(u2, ϕ)= B(u1 + u2, ϕ) = 〈βB(u1 + u2), ϕ〉,

establishing linearity of βB. Since B is continuous there exists C > 0 suchthat

‖βB(u)‖ = sup‖ϕ‖=1|〈βB(u), ϕ〉| = sup

‖ϕ‖=1|B(u, ϕ)| ≤ sup

‖ϕ‖=1C ‖u‖ ‖ϕ‖ = C ‖u‖

which implies continuity of βB. Moreover, the strictly positive definitenessof B implies θ ‖u‖2 ≤ B(u, u) = 〈βB(u), u〉, i.e., βB is strictly positive definite.Step 2: By Theorem 6.12, βB has an inverse (βB)−1 ∈ B(H,H) with operatornorm

∥∥∥β−1B

∥∥∥op≤ 1

θ. Thus the equation βB(u) = f has the unique solution

u f := β−1B ( f ) and we find∥∥∥u f

∥∥∥ =∥∥∥β−1

B ( f )∥∥∥ ≤ ∥∥∥β−1

B

∥∥∥op‖ f ‖ ≤ 1

θ‖ f ‖ .

4. Solving the weak elliptic Dirichlet boundary problem

We will first consider the solvability of the weak elliptic boundary problemin case L is a strictly positive definite operator. Then we investigate whathappens in the general case.

Theorem 6.16. If bi ≡ 0 for all 1 ≤ i ≤ n and c ≡ 0, then the weak ellipticDirichlet boundary value problem has a unique weak solution, i.e., for all

f ∈ L2(Ω), there exists exactly one u f ∈

H1,2(Ω) with B(u f , ϕ) = 〈 f , ϕ〉L2 for

all ϕ ∈

H1,2(Ω) where B is the associated bilinear form in Corollary 6.10with bi ≡ 0 for all 1 ≤ i ≤ n and c ≡ 0. Moreover, the ‘inverse’ operator

L−1 : L2(Ω)→

H1,2(Ω), L−1( f ) := u f

is compact.

For the proof, we need the following estimate.


Lemma 6.17 (Poincare inequality). Let Ω ⊆ Rn be open and bounded. Thenthere exists C > 0 such that

‖u‖L2(Ω) ≤ C∥∥∥∂xiu

∥∥∥L2(Ω)

∀ u ∈

H1,2(Ω) and ∀ 1 ≤ i ≤ n.

In particular, for ∇u := (∂x1u, . . . , ∂xnu) and |∇u|2 :=∑n

i=1(∂xiu)2 and

‖∇u‖L2(Ω) :=(∫

Ω|∇u|2

)12 , we obtain

‖u‖L2(Ω) ≤ C√

n ‖∇u‖L2(Ω)

Proof. Left as an exercise to the reader. Otherwise see for instance [Alt] or[Bressan].

In fact, Lemma 6.17 (Poincare inequality) belongs rightfully in Section 5.4before the Holder and Sobolev embedding theorems since it is needed fortheir proofs (which we skipped).

Proof of Theorem 6.16. We want to apply Theorem 6.14 (Lax-Milgram),but this needs some preparations.Step 1: The bilinear form

(6.18) B :

H1,2(Ω) ×

H1,2(Ω)→ R, B(u, ϕ) =

∫Ω

n∑i, j=1

ai j(x)∂xiu ∂x jϕ

is bounded as the following estimates will show: First consider

|B(u, ϕ)| ≤n∑

i, j=1

∫Ω

∣∣∣ai j(x)∂xiu ∂x jϕ∣∣∣ ≤ n∑

i, j=1

∥∥∥ai j

∥∥∥L∞(Ω)

∥∥∥∂xiu∥∥∥

L2

∥∥∥∂x jϕ∥∥∥

L2 .

Since∥∥∥ai j

∥∥∥L∞(Ω)

< ∞ for all 1 ≤ i, j ≤ n and ‖u‖H1,2 = ‖u‖L2 +∑n

i=1

∥∥∥∂xiu∥∥∥,

there exists C > 0 such that we get the estimate

· · · ≤ C ‖u‖H1,2 ‖ϕ‖H1,2 .

Step 2: We show that B is strictly positive definite. Lemma 6.17 (Poincareinequality) yields the existence of some C > 0 such that

‖u‖L2 ≤ C ‖∇u‖L2 ∀ u ∈

H1,2(Ω) .

By assumption, (ai j)1≤i, j≤n is elliptic with constant θ > 0 such that we get

B(u, u) =

∫Ω

n∑i, j=1

ai j(x)∂xiu ∂x ju ≥∫Ω

n∑i=1

θ ∂xiu ∂xiu = θ ‖∇u‖2L2 .

4. SOLVING THE WEAK ELLIPTIC DIRICHLET BOUNDARY PROBLEM 65

Combining both inequalities, we obtain

‖u‖H1,2 = ‖u‖L2 + ‖∇u‖L2 ≤ C ‖∇u‖L2 + ‖∇u‖L2

= (C + 1) ‖∇u‖L2 ≤C + 1θ

B(u, u).

Then θ := θ1+C > 0 is the constant of strictly positive definiteness of B.

Step 3: Since B is strictly positive definite, Theorem 6.14 (Lax-Milgram)

is valid and implies that, for all F ∈

H1,2(Ω), there exists precisely one

uF ∈

H1,2(Ω) such that B(uF , ϕ) = 〈F, ϕ〉H1,2 for all ϕ ∈

H1,2(Ω). Moreprecisely, in analogy to (6.15), we obtain, for the bilinear form in (6.18),the linear operator

βB :

H1,2(Ω)→

H1,2(Ω), βB(u) = uB

which is bounded and strictly positive definite with constant θ > 0. Its in-verse β−1

B gives us uF := β−1B (F) satisfying B(uF , ϕ) = 〈F, ϕ〉H1,2 .

Step 4: By Theorem 5.16 (Sobolev embeddings), the embedding

Id :

H1,2(Ω)→ L2(Ω)

exists and is compact. Since L2(Ω) and

H1,2(Ω) are Hilbert spaces, wecan, according to Theorem 4.7 (Riesz), identify (L2(Ω))∗ ' L2(Ω) and(

H1,2(Ω))∗'

H1,2(Ω). Thus we can see the adjoint operator

Id∗ : (L2(Ω))∗ →(

H1,2(Ω))∗

as operator Id∗ : L2(Ω)→

H1,2(Ω) .

Moreover, we get for all f ∈ L2(Ω) and all ϕ ∈

H1,2(Ω)

〈 f , Id(ϕ)〉L2 = 〈Id∗( f ), ϕ〉H1,2 = 〈βB(β−1B (Id∗( f ))), ϕ〉H1,2 = B

(β−1

B (Id∗( f )), ϕ).

The operator

L−1 := β−1B Id∗ : L2(Ω)→

H1,2(Ω)is compact due to Lemma 4.9 and u f := L−1( f ) solves B(u f , ϕ) = 〈 f , ϕ〉L2

uniquely for all ϕ ∈

H1,2(Ω). Thus Corollary 6.10 yields the claim.

APPENDIX A

Appendix

1. Basic facts from topology

We recall the following basic definition.

Definition A.1. Let X be a set. A topology T on X is a family of subsets ofX satisfying the following properties.1) X ∈ T and ∅ ∈ T .2) Let I be an arbitrary index set and Ui ∈ T for all i ∈ I. Then we also

have(⋃

i∈I Ui)∈ T .

3) Let I be a finite index set and Ui ∈ T for all i ∈ I. Then we also have(⋂i∈I Ui

)∈ T .

The tuple (X,T ) is said to be a topological space. The elements of T arereferred to as open sets and their complements in X as closed sets.

2. Basic facts concerning ordinary differential equations

We recall the following basic definition.

Definition A.2. Let D ⊆ Rn+1 be open with variables (t, x) = (t, x1, . . . , xn)and let f = ( f1, . . . , fn) : D → Rn be a function. A (system of) ordinarydifferential equation(s) (ODE) is an equation of the form

x′ = f (t, x),

i.e., a system x′1 = f1(t, x1, . . . , xn),...

......

x′n = fn(t, x1, . . . , xn).A solution of x′ = f (t, x) is a curve x = (x1, . . . , xn) : I ⊆ R → Rn

defined on an interval I ⊆ R such that (t, x(t)) | t ∈ I ⊂ D and satisfyingx′(t) = f (t, x(t)) for all t ∈ I.

Existence and uniquness of solutions are assured by

Theorem A.3 (Picard-Lindelof). If f is continuous and in addition locallylipschitz w.r.t. the x-variable, then every initial value problem of x′ = f (t, x)has a unique, maximal solution, i.e., for all (τ, ξ) in the domain of definition,

67

68 A. APPENDIX

there exists precisely one solution x with x(τ) = ξ and whose interval ofdefinition is maximal among all possible intervals of definition of solutionsof this initial value problem.

Proof. Cf. for example [Hohloch2].

Notation A.4. The unique solution for the initial value problem (τ, ξ) isoften written as x(t) = x(t, τ, ξ) and considered as function of t, τ and ξ.

We recall some basic definitions and facts.

Definition A.5. 1) An ordinary differential equation x′ = f (t, x) is said tobe autonomous if the function f does not depend on t. If f depends ont, the equation is called nonautonomous.

2) An ordinary differential equation is complete if all solutions admit R asinterval of definition.

The solutions of autonomous equations have nice properties:

Proposition A.6. Let M ⊆ Rn, f ∈ Ck(M,Rn) with k ≥ 1 and assumex′ = f (x) to be complete with solutions remaining in M. Then the map

Φ : R × M → M, Φ(t, z) := x(t, 0, z),

called the flow of x′ = f (x), is welldefined and satisfies Φ ∈ Ck(R × M,M).Usually the flow is written as Φt(x) := Φ(t, x) for all t ∈ R and all x ∈ Mand seen as map Φt : M → M. It has the following properties:

1) Φ0(z) = z ∀ z ∈ M, i.e., Φ0 = Id.2) Φs(Φt(z)) = Φs+t(z) ∀ z ∈ M, ∀ s, t ∈ R, briefly Φs+t = Φs Φt.

Proof. Cf. for example [Hohloch2].

In particular, Φt : M → M is invertible with (Φt)−1 = Φ−t for all t. More-over, the flow induces an R-action on M via

R × M → M, (t, z) 7→ Φt(z).

3. Basic facts concerning differentiable manifolds

Definition A.7. Smooth manifold; Riemannian metric; Riemannian mani-fold, Hessian, gradientstill to be written...

Definition A.8. 2-form, closed, nondegeneratestill to be written...

5. HAUSDORFF MAXIMALITY PRINCIPLE 69

4. Basic facts from measure theory

Theorem A.9. dominated convergence.still to be written...

5. Hausdorff maximality principle

We recall a few concepts and results from set theory.

Definition A.10. A set M carries a partial order or is referred to aspartially ordered by if we have for all x, y, z ∈ M1) x x,2) x y and y x implies x = y,3) x y and y z implies x z.

We are interested in particular in sequences of nested subsets and their be-haviour.

Definition A.11. Let (M,) be a partially ordered set and M ⊆ M. Thesubset M is totally ordered if any x, y ∈ M satisfy either x y or y x.A subset M ⊆ M is maximal among all totally ordered sets if M is notcontained in any other totally ordered subset of M.

The following is a variant of Zorn’s lemma or the axiom of choice.

Theorem A.12 (Hausdorff maximality principle). Let M be a partiallyordered set. Then any totally ordered subset of M is contained in a maximal,totally ordered subset of M.

Bibliography

[Adams & Fournier] Adams, Robert A.; Fournier, John J. F.: Sobolev spaces. Second edi-tion. Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press,Amsterdam, 2003. xiv+305 pp.

[Alt] Alt, Hans Wilhelm: Linear functional analysis. An application-oriented intro-duction. Translated from the German edition by Robert Nurnberg. Universitext.Springer-Verlag London, Ltd., London, 2016. xii+435 pp.

[Audin & Damian] Audin, Michele; Damian, Mihai: Morse theory and Floer homology.Translated from the 2010 French original by Reinie Erne. Universitext. Springer,London; EDP Sciences, Les Ulis, 2014. xiv+596 pp.

[Bressan] Bressan, Alberto: Lecture notes on functional analysis. With applicationsto linear partial differential equations. Graduate Studies in Mathematics, 143.American Mathematical Society, Providence, RI, 2013. xii+250 pp.

[Chacon & Rafeiro & Vallejo] Chacon, Gerardo R.; Rafeiro, Humberto; Vallejo, JuanC.: Functional analysis – a terse introduction. De Gruyter Graduate. De Gruyter,Berlin, 2017. xiii+227 pp.

[Farlow] Farlow, Stanley J.: Partial differential equations for scientists and engineers.Revised reprint of the 1982 original. Dover Publications, Inc., New York, 1993.xii+414 pp.

[Haase] Haase, Markus: Functional analysis. An elementary introduction. GraduateStudies in Mathematics, 156. American Mathematical Society, Providence, RI,2014. xviii+372 pp.

[Hohloch1] Multivariate Calculus. Lecture notes, academic year 2016-17, University ofAntwerp.

[Hohloch2] Gewone differentiaalvergelijkingen en dynamische systemen. Lecture notes,academic year 2016-17, University of Antwerp.

[Lowen] Lowen, Bob: Functionaalanalyse. Lecture notes, academic year 2016-17, Uni-versity of Antwerp.

[McDuff & Salamon] McDuff, Dusa; Salamon, Dietmar: J-holomorphic curves and sym-plectic topology. First edition. American Mathematical Society Colloquium Pub-lications, 52. American Mathematical Society, Providence, RI, 2012. xiv+726 pp.

[Milnor] Milnor, John: Morse theory. Based on lecture notes by M. Spivak and R. Wells.Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton,N.J. 1963 vi+153 pp.

[Salamon] Salamon, Dietmar: Lectures on Floer homology. Symplectic geometry andtopology (Park City, UT, 1997), 143 – 229, IAS/Park City Math. Ser., 7, Amer.Math. Soc., Providence, RI, 1999.

[Simon] Simon, Barry: Operator theory. A Comprehensive Course in Analysis, Part 4.American Mathematical Society, Providence, RI, 2015. xviii+749 pp.

71

72 BIBLIOGRAPHY

[Schwarz] Schwarz, Matthias: Morse homology. Progress in Mathematics, 111.Birkhauser Verlag, Basel, 1993. x+235 pp.

Index

σ-algebra, 32(Frechet) derivative, 43Weak topology, 28Weak* topology, 28

Adjoint operator, 37Affine hyperplane, 23Annihilator, 38Autonomous, 68

Baire, 4, 5Baire space, 4Baire theorem, 5Banach adjoint, 37Banach adjoint operator, 37Banach space, 5Banach-Alaoglu, 30Banach-Steinhaus, 9Bidual, 28Bidual space, 28Bilinear form, 62Bilinear operator, 11, 12Borel set, 32

Cauchy-Riemann operator, 58Closed, 16, 67Closed graph theorem, 17Closed operator, 16Closed set, 67Compact operator, 35Compactly contained, 18Complete, 68Continuous bilinear form, 62Convergence in norm, 27Convex, 24Critical, 1Critical point, 1

Deformed sphere, 2Dense, 3Derivative, 43Differentiable, 43Dirac operator, 58Dirichlet boundary condition, 57Domain, 15Dual, 27Dual paring, 36Dual space, 8, 27

Elliptic, 58Embedding theorem Ck+1 in Holder

spaces, 53Embedding theorem Holder in

Holder spaces, 54Embedding theorem Holder in

Sobolev spaces, 54Embedding theorem Sobolev in

Holder spaces, 55Embedding theorem Sobolev in

Sobolev spaces, 54

Flow, 68Frechet, 18Frechet differentiable, 43Frechet space, 18Fredholm, 35Fredholm alternative, 39Fredholm index, 35, 44Fredholm operator, 35Fredhom map, 44Functional, 19

Gateaux differentiable, 44Generic, 5Gradient, 68

73

74 INDEX

Holder coefficient, 53Holder constant, 52Holder functions, 53Holder number, 53Holder space, 53Hahn-Banach, 20, 22Hahn-Banach functional, 19Hahn-Banach theorem, 22Hausdorff maximality principle, 69Hellinger, 17Hellinger-Toeplitz, 17Hilbert adjoint operator, 36Homogeneous, 57Hyperplane, 23

Image, 16Implicite function theorem, 44Inverse function Theorem, 44Inverse mapping, 15Inverse mapping theorem, 15Isometric, 28

Laplace operator, 57Lax-Milgram, 62Lebesgue function, 47Lebesgue space, 29, 47Length of multi index, 48Linear operator, 8Lipschitz boundary, 51

Manifold, 1, 68Maximal, 69Maximality principle, 69Meager, 4Metric, 68Minkowski, 20Minkowski functional, 20Morse function, 2, 7Multi index, 48

Nonautonomous, 68Nondegenerate, 2Nonmeager, 4Nowhere dense, 3

ODE, 67Of first Baire category, 4Of first category, 4Of second Baire category, 4Of second category, 4

Open, 13, 67Open map, 13Open mapping, 13Open mapping theorem, 14, 15Open set, 67Operator norm, 8Ordinary differential equation, 67

Pairing, 36Partial order, 69Partially ordered, 69Picard-Lindelof, 67Poincare inequality, 64Pointwise bounded, 9Positive definite, 61Probability measure, 33Prohorov, 33

Radon, 33Range, 16Reflexive, 29Regular, 33Regular measure, 33Regular value, 44Rellich’s theorem, 51Rellich’s trace theorem, 52Riemannian manifold, 68Riemannian metric, 68Riesz, 33, 37

Sard’s theorem, 45Semi-Fredholm estimate, 40Seminorm, 18Separating, 18Separation, 24Sequentially compact, 29Sobolev functions, 49Sobolev number, 49Sobolev space, 49, 50Solution, 67Strictly convex, 25Strictly positive definite, 61, 62Strong convergence, 27Szego, 12Szego’s theorem, 12

Theorem of Banach-Alaoglu, 30Theorem of Banach-Steinhaus, 9Theorem of Hahn-Banach, 20Theorem of Lax-Milgram, 62

INDEX 75

Theorem of Picard-Lindelof, 67Theorem of Prohorov, 33Theorem of Riesz, 37Theorem of Riesz-Radon, 33Theorem of Szego, 12Toeplitz, 17Topological space, 67Topology, 67Total variation, 32Totally ordered, 69Trace operator, 52Translation invariant, 18

Uniformly bounded, 9Uniformly continuous, 8

Van der Waerden’s function, 6Variation measure, 32

Weak boundary problem, 60Weak convergence, 27, 33Weak convergence of measures, 33Weak derivative, 49Weak elliptic boundary problem, 60Weak elliptic Dirichlet boundary

problem, 60Weak limiet, 27Weak solution, 60Weak* convergence, 27Weak* limiet, 27Weakly sequentially compact, 29Weakly* sequentially compact, 29Weierstraß function, 5

Zorn’s lemma, 69

(university of antwerp, belgium)

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