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1

A quick look at topological and functionalspaces

”The unified character of mathematics lies in its verynature; indeed, mathematics is the foundation of all ex-act natural sciences.”

David Hilbert (1862-1943)

Nowadays, functional analysis, that is mainly concerned with the study ofcomplete normed vector spaces, occupies a central place in modern mathe-matical analysis. Initially motivated by the understanding and the study ofdi!erental and integral equations arising in applied mathematics, it has largelydeveloped and evolved around the theory of Banach and Hilbert spaces andtheir rich geometric structure. The importance and the versatility of Hilbertspaces is exampled by the space of Lebesgue square integrable functions. Inthis context, most functional spaces have infinite dimension and the classicaltheory focusses on linear operators between these spaces.

To better understand the conceptual breakdown in real analysis o!ered bythe new functional spaces, we introduce the following example, borrowed from[LV02]. Consider for instance the wave equation model, a simplified modeldescribing the transversal oscillations u = u(x, t) of a stretched vibratingstring in one dimension of space

!2u

!t2= c2 !2u

!x2

and supposed to be pegged at its two endpoints, i.e., edowed with the bound-ary conditions

u(0, t) = u(1, t) = 0 .

The physical interpretation suggests also to specify two initial conditions

u(x, 0) = u0(x) , and!u

!t(x, 0) = v0(x) ,

which prescribes the initial position of the string and the initial velocity ofthe points of the string. The natural setting for finding a solution with finiteenergy, i.e., such that

! 1

0

"!u

!t

#2

< +! , and! 1

0

"!u

!x

#2

< +! ,

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4 1 A quick look at topological and functional spaces

is to use Fourier series and then the solution will have the general form

u(x, t) =!$

k=1

%ak cos(k"ct) + bk sin(k"ct)

&sin(k"x) ,

and the initial and boundary conditions allow to explicit the coe"cients ak

and bk as

ak = 2! 1

0u0(x) sin(k"x)dx , and bk =

2k"c

! 1

0v0(x) sin(k"x)dx .

Such solution involves an infinite sum and is described by a denumerableset of coe"cients ak and bk. This suggests that the space of solution shall beinfinite dimensional. And this statement showed the limits of the yet known re-sults in real analysis. Indeed, the classical Bolzano-Weierstrass theorem aboutthe notion of convergence in a finite dimensional Euclidean space (i.e., everybounded sequence admits a bounded subsequence in Rn) breaks apart. Fortu-nately, Hilbert spaces were introduced and provided a convenient setting foranalyzing this type of problem.

In this chapter we summarize many of the abstract concepts, definitionsand theoretical results on the functional spaces that are relevant in functionalanalysis and important for understanding the properties of the solutions ofpartial di!erential equations. These abstract spaces, metric spaces, normedspaces, inner-product spaces, are topological algebraic spaces that have allbeen introduced in the last three decades of the nineteenth century and thefirst decades of the twentieth century1. They ultimately led to a generalizationof the notions of functions, continuity, di!erentiability and integrability. Untilthen, functions were assumed to be continuous, have derivatives at almost allpoints and were integrable by existing integration methods.

In Section 1, we recall the elementary topological spaces and the fun-damental properties, separability, compactness and completeness. Section 2,Lesbegue integration is introduced as simply as possible, without referringexplicitly to the measure theory, essentially in view of presenting Lp spaces.Hilbert spaces are the core of Section 3, in which the projection theorem andRiesz lemma are exposed. In Section 4, distributions are classically discussedin connection with functional analysis to show how this generalization of func-tions is useful for expressing solutions of partial di!erential equations. Finally,Section 5, we introduce Sobolev spaces which o!er a convenient setting forinvestigating partial di!erential equations.1 Students and persons interested in biographical notes about the ”founders” of

functional analysis will read with profit the comprehensive introduction to func-tional analysis by Karen Saxe [Sax01].


1.1 Elementary topological spaces 5

1.1 Elementary topological spaces

Before dealing with Banach and Hilbert spaces, we shall first recall some basicnotions and results from metric and topological spaces. The reader must keepin mind that important issues about partial di!erental equations concern thenotions of convergence (or limits) and continuity of functions. For instance,let consider a set of points X with a notion of distance between any two pointsof X. The convergence of a sequence of points (xn)n"1 " X to a point x # Xconsists in measuring the distance from xn to x and looking if this distancetends to 0 as n tends to infinity. There is a general setting for this concept.

1.1.1 Metric spaces

The notion of abstract metric space is due to M. Frechet2 (1878-1973).

Definition 1.1.1. A metric space is a couple (X, d), where X is a set and dis a metric or a distance function on X, i.e., d : X $X % R+ is such that

1. for any x, y # X, d(x, y) & 0 (non-negativity)2. for any x, y # X, d(x, y) = 0 if and only if x = y ( identity)3. for any x, y # X, d(x, y) = d(y, x) ( symmetry)4. for any x, y, z # X, d(x, z) ' d(x, y) + d(y, z) ( triangle inequality).

If identity 2 does not hold, then d is then called a semi-metric. Usually, onlythree conditions are used to define a distance function. Indeed, the first ofthese conditions is a property that follows from the other three, since:

2d(x, y) = d(x, y) + d(y, x) & d(x, x) = 0 .

Furthermore, the inverse triangle inequality is straightforward to obtain

|d(x, y)( d(z, y)| ' d(x, z) .

In this definition of metric space, the nature of the elements in the space isnot significant. For most problems in this textbook, metric spaces of functionswill be considered, when looking for solutions of partial di!erential equations.

Sequence spaces and function spaces

In the sequel, we will often deal with the following infinite dimensional metricspaces of real or complex sequences,

1. the metric space #! of all bounded sequences (xn)n"1, for the metric

d(x, y) = supi|xi ( yi| ;

2 M. Frechet, Sur quelques points du calcul fonctionnel, Rendic. Circ. Mat. Palermo,22, 1-74, (1906).



2. the metric space c0 of all sequences that converges to 0 with the metricof #! (as we observe that c0 " #!);

3. the metric space #p (1 ' p < !) consisting of all sequences (xn)n"1 suchthat

'!i=1 |xi|p < !, for the metric

d(x, y) =

(

)$

i"1

|xi ( yi|p*

+1/p

;

In this regard, the space #1 is the space of all absolutely convergent se-quences, i.e., (xn)n"1 is in #1 if the series

'i"1 |xi| converges. Probably

the most important among all #p-spaces is the space #2.

Let $ be any closed, bounded domain in Rn. A natural measure of the dis-crepancy between two continuous functions f and g is given by

d(f, g) = supx#!

|f(x)( g(x)| , for all x # $ . (1.1)

The subspace of all continuous functions on $ supplied with the metric (1.1)is denoted (C0($), |·|), or simply C0($). By extension, the space of all contin-uous functions on a closed, bounded domain $ whose derivatives up to orderk are continuous are denoted by Ck($). It is a metric space for the distancefunction

d(f, g) =$

|"|"k

supx#!

|D"f(x)(D"g(x)| ,

where we introduced the classical di!erential notation

D"f =!|"|f

!x"11 . . . !x"n

nwith |%| =

n$

i=1

%i . (1.2)

Let $ " Rn be a compact domain, Jordan measurable, we can consider an-other metric on C0($)

d(f, g) ="!

!|f(x)( g(x)|pdx

#1/p

, (p & 1) ,

where d defines a metric thanks to the Minkowski inequality (see Section 1.2)

"!

!|f(x) + g(x)|pdx

#1/p

'"!

!|f(x)|p

#1/p

+"!

!|g(x)|p

#1/p

, p & 1 .

The support of a function f : X % Cn, denoted by supp(f), is the closure ofall points x such that f(x) )= 0.



1.1.2 The topology of metric spaces

A specific subset of points in X containing a given point x # X defines aneighborhood of x. Let (X, d) be a metric space and r a stricly positive scalarvalue. The set Br(x) = {y # X : d(x, y) < r} is called the open ball aroundx with radius r. A point x # U " X is called an interior point of U if Ucontains some ball around x and then U is called a neighborhood of x. A pointx is called a limit point of U if (Br(x)\{x}) * U )= + for every ball around x.A point x is called an isolated point of U if there exists a neighborhood of xnot containing any other point of U . A set U is dense in M if every point ofM is a limit point of U .

The closure of U , denoted by U , is the set of points x # X such that anyopen ball Br(x) (r > 0) contains a point of U , i.e., U with its limits points.

The interior of U , denoted by$U or int(E), is the set of points x # X such

that there exists an open ball Br(x) (r > 0) which is contained in U , i.e., theset of interior points of U .

The set U is bounded if for each x # U , there exists r > 0 such thatU , Br(x). The set U is totally bounded if and only if, for any r > 0, thereexists a finite cover (Ui) of U with balls of radius r (each set Ui in the familyis of size r or less). In particular, we have the properties

1. U is open if and only if U =$U ;

2. U is closed if and only if U = U and3. U is dense in X if and only if U = X.

A set U containing only interior points is called open, it is then a union ofopen balls. Its complement is called closed. A family of sets is called a coverof U , if U is contained in the union of these sets. It is an open cover if eachset is open. If C is a cover of U , a subcover of U is a subset of C that stillcovers U . A set U is compact if its of each open cover of U contains a finitesubcover. A set U is sequentially compact if every sequence of U contains aconvergent subsequence.

Proposition 1.1.1. For any subset of the Euclidean space Rn, the followingfour conditions are equivalent

1. Every open cover has a finite subcover.2. Every sequence in the set has a convergent subsequence, the limit point of

which belongs to the set.3. Every infinite subset of the set has at least one accumulation point in the

set.4. The set is closed and bounded.

In other spaces, these conditions may or may not be equivalent, depending onthe properties of the space. The balls generate a topology on X, making it atopological space.



Definition 1.1.2. A topological space is a couple (X, T ) where X is anonempty set of points and T is a collection of open subsets of X satisfy-ing the axioms

1. The emptyset and X are in T ;2. The intersection of any finite collection of sets in T is also in T ;3. The union of any collection of sets in T is also in T .

The collection T is called a topology on X.

If X is any nonempty set, there are usually di!erent choices for the topologyT . Two interesting choices are

1. T = {+, X}, called the trivial topology ;2. T = P (X) (the power set of X) that consists of the collection of all subsets

of X, called the discrete topology.

Given two topologies T1 and T2 on X, T1 is called coarser (or weaker) than T2

if and only if T1 , T2, and T2 is then finer than T1. We consider the followinguseful topological results.

Proposition 1.1.2. Given a metric space (X, d), the following assertions hold

1. The sets + and X are both open and closed.2. A set O in (X, d) is open if and only if its complement Oc = X\O is

closed.3. Any finite intersection of open sets is open.4. Any intersection of closed sets is closed.5. The union of any finite number of closed sets is closed.6. If E is a compact subset of (X, d), then E is closed.

1.1.3 Separability, compactness and completeness

A metric space (X, d) is separable if it contains a countable dense subset, i.e.a subset with a countable number of elements whose closure is the space itself.

Recall that a sequence (xn)n"1 of elements in X is said to be Cauchy(or is called a Cauchy sequence) if given any & > 0, there exists an integern0 such that for all m & n0, d(xn, xm) ' &, whenever n. Interestingly, anyconvergent sequence is Cauchy. While in Rn the converse is true, there existsmetric spaces in which the converse does not hold.

A subset U of (X, d) is called complete if and only if every Cauchy sequencein U converges to a point of U and (X, d) is complete if and only if everyCauchy sequence converges. Hence, we deduce that a sequentially compactspace is also complete.

Lemma 1.1.1. Suppose f # C0(C) is defined on a sequentially compact met-ric space X. Then, the following assertions hold

1. the modulus of f is bounded and f attains its bound.



2. f is uniformly continuous, i.e., given & > 0, there exists ' > 0 such that

d(x, y) ' ' - |f(x)( f(y)| ' & .

We recall an important result in Euclidean and arbitrary metric spaces.

Theorem 1.1.1 (Heine-Borel).

• A subset U of Rn is compact if and only if it is closed and bounded.• A subset U of a metric space (X, d) is compact if and only if it is complete

and totally bounded.

and we can summarize

Corollary 1.1.1 (Equivalent forms of compactness). Let (X, d) be ametric space. The following properties are equivalent

1. X is sequentially compact;2. X is complete and totally bounded;3. X is compact.

Many of the metrics that are of interest for this textbook arise from norms.

1.1.4 Normed spaces

Definition 1.1.3. A normed linear space over a field K (K = R or C) is alinear space V together with a mapping . · . % R called a norm satisfying

1. for any v # V , .v. & 0, (nonnegativity)2. .v. = 0 if and only if v = 0, (nondegeneracy)3. for every v # V and ( # K, .(v. = |(| .v., (multiplicativity)4. for every v, w # V , .v + w. ' .v.+ .w., ( triangle inequality).

Norms give always rise to metrics. Indeed, if (V, . · .) is a normed space, wedefine a metric d on V by posing

d(v, w) = .v ( w. .

Hence, (V, d) is a metric space and all notions associated to metric spacesapply, notably continuity, compactness and completness. But notice that notall metrics come from norms.

Hereafter are some basic examples of normed spaces.

1. V = C with .z. = |z|.2. V = Rn with .x.2 =

,x2

1 + · · · + x2n for x = (x1, . . . , xn). This is the

Euclidean norm on Rn. Other norms on Rn can be also defined

.x.1 = |x1| + · · · + |xn| , .x.! = max1%i%n

(|xi|) .



3. Let $ be a closed, bounded subset of Rn. The function

.f. = maxx#!

|f(x)|

defines a norm on C0($).4. Let x = (xn)n"1 # #p. The function defined by

.x. =

- !$

n=1

|xn|p.1/p

(1.3)

is a norm in #p. Furthermore, we observe that the Minkowski inequalityis the triangle inequality for this norm, i.e., given (xn)n"1, (yn)n"1 # #p,

- !$

n=1

|xn + yn|p.1/p

'- !$

n=1

|xn|p.1/p

+

- !$

n=1

|yn|p.1/p

.

Let (V, .·.) be a normed space. A sequence (xn) of elements of V convergesto x # V , if for every & > 0 there exists n0 such that for every n & n0,.xn ( x. ' &. Consider the space C0($) of all continuous functions definedon a closed bounded set $ " Rn. A sequence of function (fn)n"1 is uniformlyconvergent to f if for every & > 0 there exists a constant n0 such that for allx # $ and for all n > n0, we have |fn(x) ( f(x)| ' &. Notice that the norm.f.! = maxx#! |f(x)| defines the uniform convergence and is often calledthe uniform norm.

The normed space V is imbedded in the normed space W , and we writeV % W if the following conditions are satisfied

1. V is a vector subspace of W2. the identity operator I defined on V to W by Ix = x for all x # V is

continuous.

A normed space is complete if it is complete for the corresponding metric. Acomplete normed space is called a Banach space. It can then be checked thatthe uniform norm makes C0(C) into a Banach space. If I , R is a compactinterval, then C0(I) with the maximum norm is a Banach space. Likewise, #p

is a Banach space for 1 ' p < !, for the norm defined by (1.3).

Lemma 1.1.2. If S is a closed subspace of a Banach space V and M is afinite dimensional subspace, then S + M is closed.

A linear map A between two normed spaces (V, . · .V ) and (W, . · .W ) iscalled a linear operator. The kernel and the range (or image) of A are thesets defined as usual by

Ker(A) = {f # V , Af = 0} and Im(A) = {g # W , g = Af} .

The operator A is called bounded if the operator norm



.A. = sup&f&V =1

.Af.W

is finite. The set of all bounded linear operators from V to W is denoted byL(V,W ) or L(V ) if V = W . It is a normed space with the operator norm anda Banach space if W is a Banach space.

Lemma 1.1.3. An operator A is bounded if and only if it is continuous.

An operator in L(V, C) is called a bounded linear functional and the spaceV ' = L(V, C) is called the dual space of V . The space V ' is sometimes denotedby V (.

Since many norms arise from inner product, we shall now define this notion.

1.1.5 Inner product spaces

Definition 1.1.4. Let V be a linear vector space on K. An inner product onV is a mapping (·, ·) : V $ V % K satisfying the conditions

1. (v, v) & 0 for all v # V ; (nonnegativity)2. (v, v) = 0 if and only if v = 0; (nondegeneracy)3. ((v, w) = ((v, w) for all v, w # V and ( # K; (muliplicativity)4. (v, w) = (w, v) for all v, w # V ; (Hermitian symmetry)5. (v, w + u) = (v, w) + (v, u) for all u, v, w # V . (distributivity)

The inner product is sometimes denoted by /·, ·0 and may be called a scalarproduct. A vector space with an inner product is called an inner product spaceor a pre-Hilbert space. Inner products always allow to define norms. Indeed, if(V, (·, ·) is an inner product space, we define a norm . ·. on V by posing

.v. =,

(v, v) .

Hence, (V, . ·. ) is a normed space.Hermitian symmetry and linearity in the first variable give

(v, (w) = ((w, v) = ((w, v) = ((v, w)

as well as

(v, w + u) = (w + u, v) = (w, v) + (u, v) = (v, w) + (v, u) ,

and thus an inner product is a sesquilinear form. Subsequently, an in-ner produt on a real vector space is a positive-definite symmetric bilinearform. Clearly, the complex linear space Cn with the usual inner product(z, w) =

'ni=1 ziwi is an inner product space.

Lemma 1.1.4 (Cauchy-Schwarz inequality). If ((V, (·, ·)) is an innerproduct space, then for all v, w # V we have

|(v, w)| ' .v. .w. .



It is easy to see that the equality only occurs if v and w are colinear.

Lemma 1.1.5 (Jordan-von Neumann). A norm is associated with a innerproduct if and only if the parallelogram rule holds, i.e.,

2.v.2 + 2.w.2 = .v + w.2 + .v ( w.2

Strong and weak convergence

Since every inner product space is a normed space for the naturally definednorm .x. =

,(x, x), the notion of convergence is well defined.

A sequence of vectors (xn)n"1 in an inner product space V is said toconverge strongly (or to converge in the norm) to a vector v in V if .xn(x. %0 as n % !. A sequence of vectors (xn)n"1 in an inner product space V issaid to converge weakly to a vector v in V if (xn, y) % (x, y) as n % !, forall y in V .

Weakly convergent sequences are bounded, i.e., there exists M > 0 suchthat .xn. ' M for all n. The notion of weak convergence defines a topology onV that is called the weak topology on V . From Cauchy-Schwarzs inequality,we can deduce that the weak topology is weaker than the norm topology.Hence, a strongly convergent sequence is also weakly convergent to the samelimit, while the converse is not true in general. However, if (xn, x) %)x, x)and .xn. % .x., then we have .xn ( x. % 0 as n %!.

Complete inner product spaces

A complete inner product space is called a Hilbert space. Next are some in-teresting inner product and Hilbert spaces

1. if we are considering continuous functions defined over C, C0(I), whereI = [a, b] is a compact subset of C, is an inner product space endowedwith the inner product

(f, g) =! b

af(x)g(x)dx

and then the induced norm is then

.f.2 =

-! b

a|f(x)|2dx

.1/2

,

and not the supremum norm .f.! = supx#I |f(x)|.2. the space #2 of complex valued sequences x = (xn)n"1 is a Hilbert space

endowed with the inner product

(x, y) =!$

n=1

xnyn .


1.2 The Lebesgue integral 13

A vector v in a vector space V is called normalized or unit vector if .v. = 1.Two vectors v, w in a vector space V are orthogonal (v 1 w) if (v, w) = 0 andparallel if one is a multiple of the other. If v and w are orthogonal, we havethe Pythagorean formula

.v + w.2 = .v.2 + .w.2 . (1.4)

For v and w )= 0 in an inner product space, the projection of v on w is thevector

v =(v, w).w.2 w , or, if .w. = 1 v = (v, w)w .

We recall that a set C in a vector space is convex if for any x, y # C, we havetx + (1( t)t # C, with t # [0, 1].

Lemma 1.1.6. Let C be a nonempty convex and complete linear subspace of ainner product space V . If v # V then there exists a unique u # C minimizing.v ( u. and called the closest point (or best approximation) to v from C.Furthermore, we have v ( u 1 C, i.e., (v ( u, w) = 0 for any w # C.

Given an inner product space V and M " V , we define the orthogonal spaceof M by

M) = {v # V ; (v, m) = 0 for all m # M} .

and this space M) is often referred to as M -perp.

Lemma 1.1.7. Suppose V is an inner product space and M a complete linearsubspace of V . Then, V = M 2M), the direct sum being orthogonal.

Proposition 1.1.3. Suppose V is an inner product space and M " V . ThenM) is a linear subspace of V , M 1 M) and the intersection M * M) iseither {0} or the emptyset.

Before examining in more detail the properties of Hilbert spaces, we liketo introduce the Lesbesgue integral, a fundamental concept for understandingmost applications Hilbert space theory.

1.2 The Lebesgue integral

We shall recall that one purpose of this textbook is to introduce the mainnumerical methods for solving partial di!erential equations. Since the solu-tions to these equations involve Hilbert spaces, it is important to show thatall di!erentiable functions defined on a compact interval I = [a, b] belong toa Hilbert space endowed with the inner product

(f, g) =! b

af(x)g(x)dx .



And precisely, the smallest of such spaces is the space of Lebesgue squareintegrable functions on I. Hence, we will briefly review now this concept ofLebesgue integral3.

1.2.1 Lebesgue integration

The classical Riemann integral is suitable for dealing with continuous func-tions defined on bounded subsets of the Euclidean space Rn, or functions witha limited number of discontinuities. However, it cannot handle discontinuousfunctions. The general setting of the measure theory is suitable for resolvingthese drawbacks, the integral is then defined from the notion of size in someset V .

Note to the reader.

In this textbook, we are mainly concerned with integration and not measure.Moreover, a complete description of the measure theory is beyond the scope ofthis book, artly dedicated to undergraduate students. We have thus decided tointroduce the Lebesgue integral without referring explicitly to concepts likemeasure, following a more direct approach pionnered by [DM90]. A readerinterested in this topic will consult with profit the books listed in appendixof this chapter. Hence, graduate students and mathematicians could quietlyskip this section.

A real valued f defined on R is called a step function if it can be writtenas a finite linear combination of characteristic functions of semi-open intervalsAi = [ai, bi[, R, i.e.,

f(x) =n$

k=1

%k)Ak(x) , for all x # R , (1.5)

where %k # R and )A is the characteristic function of A, such that )Ak(x) = 1if x # [ak, bk[ and )Ak(x) = 0 otherwise. In this setting, the intervals Ak

are assumed to be disjoint and if we consider besides a minimal number ofintervals, then the representation of f is unique4. It enjoys several properties,notably

1. if f, g are step functions, then f + g and fg are step functions,2. if f is a step function and % # R, then %f is a step function,3. if f is a step function, then |f | is a step function.4. if f, g are step functions, min(f, g) and max(f, g) are step functions.3 Named after the French mathematician Henri Lebesgue (1875-1941) who intro-

duced the theory of integration in his doctoral dissertation, Integrale, longueur,aire, University of Nancy, (1902).

4 Advanced readers will easily compare this definition with the notion of a measur-able simple function having a finite range, in the measure theory.



The collection of all step functions is a vector space on R. The derivative ofa step function is the Dirac delta function '(x) = 0, if x )= 0 and '(x) = +!otherwise.

We define the integral of a step function as the Riemann integral of thisfunction, i.e., !

f =n$

k=1

%k(bk ( ak) =! !

*!f(x)dx .

And we observe that this definition is independent of any particular represen-tation of f . We have then

Lemma 1.2.1. Given f a step function whose support is contained in/n

k=1 Ak,where the Ak are disjoint semi-open intervals [ak, bk[. If, for any M > 0|f | < M , then !

|f | ' Mn$

k=1

(bk ( ak) .

Next, we give two results that will be helpful for defining the Lebesgue integral.

Theorem 1.2.1. 1. Let (fn)n"1 be a non-increasing sequence of nonnegativestep functions such that limn+! fn(x) = 0 for every x # R. Then,

limn+!

!fn(x)dx = 0 .

2. Let (fn)n"1 and (gn)n"1 be two non decreasing sequences of step functions.If limn+! fn(x) ' limn+! gn(x) for every x # R, then

limn+!

!gn(x)dx ' lim

n+!

!fn(x)dx .

And finally, we introduce the expected definition of the Lebesgue integral.

Definition 1.2.1 (Lebesgue integral). A real-valued function f defined onR is called Lebesgue integrable if there exists a sequence of step functions(fn)n"1 satisfying the following axioms

1.!$

n=1

!|fn(x)|dx < !;

2. f(x) =!$

n=1

fn(x), for every x # R such that!$

n=1

|fn(x)| < !.

The integral of f is then defined by!

f(x)dx =!$

n=1

!fn(x)dx .



In this definition, condition 2 shows that f is equal to the sum of series atpoints where the series converges absolutely. We will show that the set of allpoints where f does not coincide with

'!n=1 fn(x) is a small set, called a

null set. Hence, the series converges to f(x) for all x except a null set. Thisintroduces the concept of convergence almost everywhere.

The space of all Lebesgue integrable functions defined on R is denoted byL1(R). And we will observe that all Riemann integrable functions are Lebesgueintegrable. The space L1(R) is a vector space and the function

0is a linear

functional on L1(R).

Lemma 1.2.2. If f # L1(R) then |f | # L1(R) and we have1111!

f(x)dx

1111 '!

|f(x)|dx .

Corollary 1.2.1. Given f, g # L1(R),

1. if f =!$

n=1

fn, then!

|f(x)| dx '!$

n=1

!|fn(x)| dx;

2. min(f, g) # L1(R) and max(f, g) # L1(R).

And we have the following result.

Theorem 1.2.2. If f # L1(R), then for t scalar we have

limt+0

!|f(x + t)( f(x)|dx = 0 .

Proof. If f is a step function, the results is immediate. Suppose f is an arbi-trary Lebesgue integrable function. Then, given & > 0, if f =

'!n>0 fn there

exists n0 such that'!

n>n0

0|fn| < &/3, and we have

!|f(x + t)( f(x)|dx '

! 11111

n0$

n=1

fn(x + t)(n0$

n=1

fn(x)

11111 dx

+$

n>n0

!|fn(x + t)|dx +

$

n>n0

!|fn(x)|dx

=! 11111

n0$

n=1

fn(x + t)(n0$

n=1

fn(x)

11111 dx

+ 2$

n>n0

!|fn(x)|dx

<

! 11111

n0$

n=1

fn(x + t)(n0$

n=1

fn(x)

11111 dx + 2&/3 .

As we know that'n0

n=1 fn is a step function, we can then deduce that



limt+0

! 11111

n0$

n=1

fn(x + t)(n0$

n=1

fn(x)

11111 dx = 0 ,

and thus0|f(x + t)( f(x)|dx < & for t su"ciently small. 34

Interestingly, L1(R) can be considered as a Banach space, under specific as-sumptions, and norms can be defined. To this end, we propose now a definitionfor the null function previously evoked.

Definition 1.2.2 (null function and null set).

1. An integrable function f is called a null function if0|f | = 0. Furthermore,

two integrable functions f, g are equivalent if f ( g is a null function.2. A set X , R is called a null set or measure-zero set, if its characteristic

function is a null function.

Under this definition, any countable set if a null set and a countable unionof null sets is also a null set. For example, the set of rational numbers Q is anull set with respect to Rn, despite being dense in Rn. All subsets of Rn ofdimension smaller than n are null sets in Rn. A classical example of a null setwhich is not countable is the Cantor set [Hal50]. A set is considered null if itis a subset of a null set.

1.2.2 Notions of convergence

In measure theory, a property hold almost everywhere, abbreviated a.e., if theset of elements for which this property is not satisfied is a null set. Hence, iff, g # L1(R) and if the set of elements x # R for which f(x) )= g(x) is a nullset, then f equals g almost everywhere, i.e., f = g a.e.

At this point, we can introduce the equivalence class of f # L1(R) as theset of all functions g # L1(R) which are equivalent to f , i.e.,

[f ] = {g # L1(R) ,

!|f ( g| = 0} ,

and then consider the space L1(R) of all equivalence classes of Lebesgue inte-grable functions, endowed with the norm

.[f ]. =!

|f | .

Then, the space (L1(R), . ·. ) is a normed space. We recall the definition ofconvergence in a normed space.

Definition 1.2.3. A sequence of functions (fn) # L1(R) converges in normto f # L1(R), and we denote by fn % f i.n., if .fn % f. % 0.



According to this definition, if fn % f i.n., then |fn|%| f | i.n. and0

fn %0

f .A sequence of function (fn) # L1(R) converges to a function f # L1(R) almosteverywhere, denoted by fn % f a.e., if fn(x) % f(x) for every x except a nullset.

We introduce two important results regarding L1(R) spaces. It relates thenotions of convergence in norm and convergence almost everywhere.

Theorem 1.2.3 (Riesz).

1. The space L1(R) is complete;2. Given a sequence (fn). If fn % f in norm, then there exists a subsequence

(fpn) of (fn) such that fpn % f a.e.

It comes directly that the limit with respect to the convergence in norm canbe interchanged with the integration, thus leading to write

!lim

n+!fn = lim

n+!

!fn .

And we shall notice here that this property does not hold when considering theconvergence almost everywhere. The next results illustrate the main di!erencebetween Lebesgue integration and other integrations.

Theorem 1.2.4 (Lebesgue’s monotone convergence). Let (fn) be a non-decreasing sequence of nonnegative integrable functions, i.e., such that for ev-ery k & 1 ,

0 ' fk(x) ' fk+1(x) , for almost every x # R .

Let f be defined as the pointwise limit of the sequence, f(x) = limn+! fn(x).Then f is integrable and

limn+!

"!fn

#=

!f .

Theorem 1.2.5 (Lebesgue’s dominated convergence). Let (fn) be a se-quence of square integrable functions converging almost everywhere to a func-tion f . Moroever, suppose there exists a square integrable function g such that|fn| ' g for all n. Then f is integrable and fn % f i.n., i.e.,

limn+!

"!fn

#=

!f .

Lemma 1.2.3 (Fatou’s lemma). Let (fn) be a sequence of nonnegative in-tegrable functions and let f = lim infn+! fn. Then

!f ' lim inf

n+!

"!fn

#.



1.2.3 Locally integrable functions

Until now, we have dealt with the integration over the whole set R, where theintegral

0f was meant for

0 +!*! f . However, we need to define the integration

over bounded intervals. Let I = [a, b] be an interval and f : I % R be afunction. We denote by

0 ba f or

0I f the integral of f over the interval [a, b]. It

corresponds to the value of the integral of the product0

f)[a,b], where )[a,b]

represents the characteristic function of [a, b]. According to this definition,0 ba f corresponds to

0f on [a, b] and zero otherwise. In addition, we have the

following conventions! b

1f = (

! a

bf and

! a

af = 0 .

We observe that if f # L1(R), then for any interval [a, b] on R,0 b

a f exists.However, the converse may not hold.

Definition 1.2.4. A locally integrable function is a function f defined on Rsuch that for any compact interval [a, b], the integral

0 ba f exists.

Under this definition, the space L1(R) is a subspace of the space of locallyintegrable functions that forms a vector space.

Lemma 1.2.4. Suppose f is a locally integrable function such that |f | ' g forsome function g # L1(R). Then, f # L1(R).

1.2.4 Lebesgue vs. Riemann integration

In this section, we assume the reader to be familiar with the definition ofthe Riemann integral and its properties. We briefly summarize this notion inorder to introduce notations.

We consider a bounded function f defined on the closed, bounded interval[a, b] " R. Suppose that a = x0 < x1 < · · · < xn = b is a partition of [a, b]together with a finite sequence of real t1, . . . , tn such that for each k & 1,xk*1 ' tk ' xk. The lower Riemann sum and upper Riemann sum of f withrespect to the partition (xn) are respectively defined by

Ln(f) =n$

k=1

mk(xk ( xk*1) , and Un(f) =n$

k=1

Mk(xk ( xk*1) ,

where mk = inf(f(x);x # [xk*1, xk]) and Mk = sup(f(x);x # [xk*1, xk]).A bounded function f defined on [a, b] is called Riemann integrable if

Ln(f) = Un(f) and we denote the Riemann integral by! b

af(x)dx .



An interesting property is that the Riemann integral is considered as the limitof the Riemann sums of a function when the size of the partition (xn) tendsto zero.

The definition of the Lebesgue integral cannot be seen as a generalizationof the Riemann integral, but the following result is interesting in this respect.

Theorem 1.2.6. If f is a Riemann integrable function on [a, b] then f isLebesgue integrable on [a, b] and both integrals coincide.

Proof. For every integer n, we consider a partition of [a, b] into 2n subintervalseach of length (b( a)/2n. Next, we define

gn(x) =2n$

k=1

mk)[xk!1*xk[(x) , and hn(x) =2n$

k=1

Mk)[xk!1*xk[(x) ,

and observe that (gn)n"1 is an increasing sequence while (hn)n"1 is a de-creasing sequence. Denoting g = limn+! gn(x) and h = limn+! hn(x) leadsto conclude that g and h are Lebesgue integrable functions such that, foralmost every x in [a, b]

g(x) ' f(x) ' h(x) .

It is not di!uclt to deduce that, for almost all x in [a, b]

limn+!

(hn(x)( gn(x)) = h(x)( g(x) .

Hence, thanks to the monotone convergence theorem, we have

0 '!

(h( g) = limn+!

!(hn ( gn) = lim

n+!

!hn ( lim

n+!

!gn

= limn+!

Un(f)( limn+!

Ln(f) = 0 .

And it follows that g = h a.e., thus f is Lebesgue integrable. Moreover! b

af(x)dx = lim

n+!

! b

agn(x)dx =

! b

ag =

! b

af ,

where the integral on the left denotes the Riemann integral and the integralon the right denotes the Lebesgue integral. 34

A version of the fundamental teorem for calculus can be given for the Lebesgueintegral.

Theorem 1.2.7 (Fundamental theorem of calculus). If f is Lebesgueintegrable on [a, b] and if we define

F (x) =! x

af

then the function F is continuous in [a, b] and di!erentiable a.e. Furthermore,F is di!erentiable a.e. and F '(x) = f(x) for almost every x in [a, b].



From this theorem, we can also deduce that the limits

F (a) = limx+a+

F (x) and F (a) = limx+b!

F (x)

exists and are finite.The following result allows to introduce a change of variable in the

Lebesgue integral in a similar way as it is performed with the Riemann inte-gral.

Lemma 1.2.5 (Change of variables). Let g be a nondecreasing di!eren-tiable function defined on a bounded interval [a, b] such that g' is integrableover [a, b]. We pose

g(a) = limx+a!

g(x) and g(b) = limx+b+

g(x) .

Suppose f is an integrable function over [g(a), g(b)]. Then, the product (f 5g)g'is integrable over [a, b] and we have

! g(b)

g(a)f(t) dt =

! b

a(f 5 g)(t)g'(t) dt .

1.2.5 The Lebesgue measure on Euclidean space

Now, we are in good condition for defining more general concepts like measuresets and the Lebesgue measure. The later is a classical manner of assigninglength, area and volume to subsets of the Euclidean space. But remember thatnot all sets are measurable.

Definition 1.2.5. We introduce the notions of measurable set and of measureas follows

1. a set A is called measurable if the characteristic function of A is a locallyintegrable function;

2. given a measurable set A. If the characteristic function )A is an integrablefunction, then the measure µ(A) of A is defined by

µ(A) =!

)A ,

and we assign µ(A) = ! if )A is not integrable.

Null sets are zero-measure sets, this justify the terminology. More generally,mesure µ is a countably additive nonnegative function µ.

Proposition 1.2.1. Let (Ak)k"1 be a sequence of disjoint measureable sets.Then A =

/k"1 Ak is measurable and we have

µ(A) = µ2 3

k"1

Ak

4=

$

k"1

µ(Ak) .



The integral over any measurable set $ can be defined by!

!f =

!f )! .

Definition 1.2.6 (Measurable function). A function f is called measur-able if there exists a sequence of step function (fn)n"1 such that fn % fa.e.

Under this definition, every integrable function is measurable. Furthermore,every locally integrable function is measurable. The measurable functions forma vector space and the following properties hold

1. if f is measurable, then |f | is measurable;2. if f, g are measurable, then f + g, fg are measurable;3. if (fn)n"1 is a sequence of measurable functions, then the functions

(inf fn)(x) = infn"1

fn(x) and (sup fn)(x) = supk"1

fn(x)

(lim inf fn)(x) = supj"l

( infn"j

fn(x)) and (lim sup fn)(x) = infj"l

(supn"j

fn(x))

are measurable functions.

1.2.6 More Lebesgue spaces

It is possible to define arbitrary measure spaces and we will now introduce afew other measure spaces.

The space L2(R)

The space of all locally integrable functions f such that |f |2 # L1(R) is de-noted by L2(R). Functions in L2(R) are also called square integrable functions.The space L2(R) is a vector space and the product of two square integrablefunctions is also a function of L2(R). Furthermore, if we consider the normdefined by .f. =

%0|f |2

&1/2, then (L2(R), . ·. ) is a complete normed space.The space of square integrable functions that vanish outside an interval [a, b]is denoted by L2([a, b]).

The spaces L1(Rn) and L2(Rn)

We like to consider the Euclidean space Rn as the measure space. Given(ak)1%k%n and (bk)1%k%n in Rn, with each ak ' bk, we consider subsets of Rn

of the form I = [a1, b1]$ · · ·$ [an, bn]. For a subset I, we define

m(I) =n5

k=1

(bk ( ak) .



And we observe that m(I) represents the length, area and volume of I whenn = 1, 2 or 3, respectively.

We have introduced the notion of step functions as real-valued functionsthat have only a finite number of elements in their range (cf. Definition 1.5).Every such function can be decomposed as a linear combination of character-istic functions. A step function is measurable if and only if each set Ai is ameasurable set. Such a function is then called a simple measurable function.Every function defined on Rn can be approximated by simple functions. Fora step function f , we write

f =n$

k=1

%k)Ik ,

and we define !f =

n$

k=1

%km(Ik) .

Next we introduce the notion of Lebesgue integrable function, that expandsnaturally Definition 1.2.1.

Definition 1.2.7 (Lebesgue integrable function on Rn). A real (or com-plex) valued function f defined on Rn is called Lebesgue integrable if thereexists a sequence of step functions (fn)n"1 satisfying the followings axioms

1.!$

n=1

!|fn| < !;

2. f(x) =!$

n=1

fn(x) for every x # Rn such that!$

n=1

|fn(x)| < !.

The integral of f is then defined by!

f =!$

n=1

!fn .

The space of all Lebesgue integrable functions on Rn is denoted by L1(Rn).Likewise, we extend Definition 1.2.4 as follows.

Definition 1.2.8. A function f defined on Rn is locally integrable, if forevery bounded interval I the product f)I is an integrable function.

By analogy with the previous section, a set A , Rn is called measurable if thecharacteristic function of A is a locally integrable function. The measure µ(A)of A is then defined by the value of the integral µ(A) =

0)A and µ(A) = !

is )A is locally integrable but not integrable. Finally, a function f defined onRn is measurable is there exists a sequence of step functions (fn)n"1 such thatfn % f a.e.



The integral over a measurable set $ , Rn is defined by the integral ofthe function f on $ and 0 everywhere else,

!

!f =

!f)! .

The space of locally integrable functions f defined on Rn such that |f |2is integrable is denoted by L2(Rn). Functions in L2(Rn) are called squareintegrable, The space of square integrable functions that vanish outside ansubset $ , Rn is denoted by L2($).

The spaces Lp(Rn)

Finally, for a bounded subset $ , Rn, we consider the space Lp($), for1 ' p < !, of all real-valued Lebesgue measurable functions defined on $such that !

!|f |p < ! .

And we define, for p = !, the space L!($) of all real-valued Lebesguemeasurable functions that are essentially bounded, i.e., ess supx#! |f(x)| < !,where

ess supx#!

|f(x)| = inf{M & 0 , |f(x)| ' M, a.e. in $} .

We observe that the spaces Lp form a sequence of embedded spaces, i.e.,

L!($) " · · · " Lp($) " · · · " L2($) " L1($) .

Actually, the space Lp consists of equivalence classes of functions, wheretwo functions belong to the same equivalence class if they coincide almosteverywhere. For 1 ' p < !, the space Lp is a linear space. For any 1 ' p < !there exists a unique q such that

1p

+1q

= 1 , with q = ! , if p = 1 .

The number q is called the Holder conjugate of p.

Lemma 1.2.6 (Holder’s inequality). Suppose 1 < p < ! and 1 < q < !are Holder conjugates. Given f # Lp($) and g # Lq($) then fg # L1($) and

.fg.L1(!) =!

!|fg| ' .f.Lp(!) .g.Lq(!) ,

1p

+1q

= 1 .

Theorem 1.2.8. For 1 ' p < !, the space Lp($) is a normed linear space,with the norm defined by

.f.Lp(!) ="!

!|f |p

#1/p

and .f.L"(!). = ess supx#!

|f(x)| . (1.6)


1.3 Hilbert spaces 25

In particular, the triangle inequality holds

.f + g.Lp(!) ' .f.Lp(!) + .g.Lp(!) ,

that is called Minkowski’s inequality in this context. Furthermore, a normedlinear space (V, . ·. ) is complete if and only if

'j"1 fj converges in norm

whenever'

j"1 .fj. converges. According to this result, the space Lp($) iscomplete and thus is a Banach space for the norm . ·. Lp(!), for any value p.

The most important Lp spaces for our purposes are the spaces where themeasure µ is the Lebesgue measure on some subset $ of Rn or µ is a countingmeasure on N. In this case, we have already introduced these spaces, denotedby #p, as the spaces of all bounded sequences (xn)n"1 satisfying

$

n"1

|xn|p < ! , with .(xn)n"1.p = (|xn|p)1/p .

We close this section on Lebesgue functions by giving an interesting result.

Theorem 1.2.9. The step functions are dense in Lp($), for each 1 ' p < !.

It remains to be stated that the space L2($) is a Hilbert space and itsnorm is induced by an inner product. This can be seen by writing

(f, g) =!

!f g .

This space will be discussed in the next section.

1.3 Hilbert spaces

The work of David Hilbert (1862-1943) on quadratic forms in infinitely manyvariables in his study of integral equations impelled the theory of Hilbertspaces. Their importance was first recognized years later by John von Neu-mann (1903-1957) in his work on unbounded Hermitian operators and he iscredited for having developed the modern theory of Hilbert spaces.

The space L2([a, b]) is a Hilbert space. We have seen that L2([a, b]) is anormed space, then we still have to show that it is complete. Let (fn)n"1 bea Cauchy sequence in L2([a, b]), we have

! b

a|fm ( fn|2 % 0 as m, n %! .

Cauchy-Schwarz’s inequality yields

! b

a|fm ( fn| '

-! b

a1

! b

a|fm ( fn|2

.1/2

=6

b( a

-! b

a|fm ( fn|2

.1/2



and the right-hand side term tends to 0 as m, n % 0. Hence, (fn)n"1 is aCauchy sequence in L1([a, b]) that converges to a function f # L1([a, b]). Thismeans that ! b

a|f ( fn|% 0 as n %! .

Thanks to the Riesz theorem, there exists a subsequence (fpn) convergent tof a.e. For any &, we obtain, by letting n %! and for pm > pn

! b

a|fpn ( f |2 ' & ,

by Fatou’s lemma. And thus f # L2([a, b]). Furthermore, we write! b

a|f ( fn|2 '

! b

a|f ( fpn |2 +

! b

a|fpn ( fn|2 < 2& ,

for n su"ciently large, and the completeness is achieved.

1.3.1 Orthonormal bases

A basis of a vector space E is a linearly independent subset B of E spanningE, i.e., such that any vector x # E can be written as x =

'nk=1 %kxn, where

xk # B and the %k are scalars. In inner product spaces, the reasons for whichbases are so important are twofold. Infinite sums are considered instead offinite linear combinations and the notion of orthogonality replaces the linearindependence property.

Let V be a Hilbert space V endowed with an inner product (·, ·). A se-quence (or a set of vectors) (vn)n"1 is called an orthonormal sequence if(vk, vj) = 'k,j , for 1 ' k, j < !, where 'jk denotes the Kronecker deltasymbol, i.e. 'jk equals one if j = k and zero otherwise. If the sequence isinfinite, then it converges weakly to 0.

For example, the set of functions fn(x) = exp(inx)/6

2", for n # Z is anorthonormal sequence for the space L2([(", "]) endowed with the L2 innerproduct

(f, g) =! #

*#f(x)g(x) dx .

Likewise, the set of Legendre polynomials Pn(x) defined by the Rodriguesformula

P0(x) = 1 , Pn(x) =6

2n + 12

12nn!

dn

dxn

%(x2 ( 1)n

&, n & 1 ,

forms an orthonormal system in the space L2([(1, 1])The next result generalizes the Pythagorean formula (1.4). It can be es-

tablished by induction.



Lemma 1.3.1 (Pythagorean formula). Let (fk)1%k%n be a set of orthog-onal vectors in an inner product space V . Then,

77777

n$

k=1

fk

77777

2

=n$

k=1

.fk.2 .

Suppose (fn)n"1 is an orthonormal sequence in an inner product space V .Then, an interesting question would be to find (complex) numbers %n suchthat, for all f # V ,

f =!$

n=1

%nfn .

Unfortunately, this cannot be achieved in general. Nevertheless, we have thefollowing results.

Lemma 1.3.2. Suppose that (fn)n"1 is an orthonormal sequence in an innerproduct space V and that f =

'!n=1 %nfn. Then, %n = (f, fn) for each n.

We call the term'!

n=1(f, fn)fn the Fourier series of f with respect to theorthonormal sequence (fn)n"1 and (f, fn) are the Fourier coe"cients of fwith respect to (fn)n"1. The following result gives details on the size of thesecoe"cients.

Theorem 1.3.1 (Bessel’s inequality). Suppose that (fn)n"1 is an or-thonormal sequence in an inner product space V . Then, for every f # Vwe have

!$

n=1

|(f, fn)|2 ' .f.2 . (1.7)

In particular, this inequality shows that the series of nonnegative numbers'!n=1 |(f, fn)|2 converges for every f # V . This property means that the

sequence (f, fn)n"1 is an element of the Hilbert space #2 of square-summablesequences.

Let (fn)n"1 be an orthonormal sequence in V . If for any f # V , thereexists coe"cients %n such that f =

'!k=1 %nfn, then the sequence (fn)n"1

is called a complete orthonormal sequence in V or an orthonormal basis forV . According to this definition, an orthonormal sequence (fn)n"1 in a Hilbertspace V is complete if for every f # V we have

f =!$

n=1

(f, fn)fn .

Actually, this equality means that

limn+!

77777f (!$

n=1

(f, fn)fn

77777 = 0 ,

with respect to the norm . ·. in V .



Theorem 1.3.2 (Parseval’s theorem). Suppose that (fn)n"1 is an or-thonormal sequence in an inner product space V . Then, (fn)n"1 is a completeorthonormal sequence if and only if for every f # V we have

!$

n=1

|(f, fn)|2 = .f.2 . (1.8)

The completeness of V is in general su"cient to ensure the convergence of theseries

'!n=1 |(f, fn)|2 as stated next.

Theorem 1.3.3. Let (fn)n"1 be a complete orthonormal sequence in a Hilbertspace V and let (%n)n"1 be a sequence of real or complex numbers. Then, theseries

'!n=1 %nfn converges if and only if

'!n=1 |%n|2 < !. Moreover, in that

case we have 77777

!$

n=1

%nfn

77777

2

=!$

n=1

|%n|2 .

And we have an important characterization of complete orthonormal se-quences.

Lemma 1.3.3. An orthonormal sequence (fn)n"1 in a Hilbert space V iscomplete if and only if

(f, fn) = 0 , for all n & 1 - f = 0 .

A Hilbert space is called separable if it contains a complete orthonormal se-quence. Finite dimensional Hilbert spaces are separable.

If V is separable, the construction of an orthonormal basis is easy. Indeed,there exists a countable total set (fn)n"1. The Gram-Schmidt orthogonal-ization procedure (cf. Section 4.2.5) allows to construct an orthonormal set(un)n"1 such that span(un) = span(fn) for any n.

Theorem 1.3.4. Every separable inner product space has a countable or-thonormal basis.

Corollary 1.3.1. If V is separable, then every orthonormal basis is countable.

A bijective linear operator A # L(V,W ), where V and W are Hilbert spaces,is called unitary if A preserves the inner products (or the norms)

(Ag, Af) = (g, f) , g, f # V .

Hence, V and W are called unitarily equivalent. Let V be an infinite di-mensional Hilbert space, and let (fn)n"1 be any orthogonal basis. The mapA : V % #2(N), f 7% ((fk, f))k"1 is unitary.

Lemma 1.3.4. Any separable infinite dimensional Hilbert space is unitarilyequivalent to #2(N).



1.3.2 The projection theorem and Riesz lemma

In this section, we consider closed vector subpaces of a Hilbert space V , thatare Hilbert spaces since a closed subspace of a complete normed space iscomplete.

Let S be a nonempty subset of a Hilbert space V . We recall that u # H isorthogonal to S if (u, v) = 0 for every v # S. Then, the orthogonal complementof S is the set of all elements in V orthogonal to S, denoted by S). Bycontinuity of the inner product, it follows that S) is a closed linear subspaceand by linearity that span(S)

)= S). Obviously, {0}) = V and V ) = {0}.

Moreover, is S is a closed subspace of V , we have S)) = S.

Lemma 1.3.5. Let S be a subspace of a Hilbert space V . Then S is dense ifand only if S) = {0}.

A fundamental property of Hilbert spaces is that the distance of a pointto a closed convex set if always attained. This result is especially importantin the approximation theory.

Theorem 1.3.5 (Closest point). Let C be a closed convex subset of aHilbert space V . For every element g # V , there exists a unique closest el-ement (or a best approximation) f # C to g minimizing .g ( f., i.e. suchthat

.g ( f. = infh#C

.g ( h. .

Proof. We prove first the existence of such point. Let (fn)n"1 be a sequencein C such that

limn+!

.g ( fn. = infh#C

.g ( h. .

Posing d = infh#C .g ( fn., we know that 12 (fm + fn) # C and thus we have

.g ( 12(fm + fn). & d , for all m, n & 1 .

From the parallelogram formula, we show that

.|fm ( fn.2 = 2.g ( fm.2 + 2.g ( fn.2 ( 4.g ( 12(fm ( fn).2 ,

and since 2.g( fm.2 +2.g( fn.2 tends to 4d2 when n, m %!, we concludethat .fm ( fn.2 tends to 0 when m, n % ! and thus, (fn)n"1 is a Cauchysequence. The limit f = limn+! fn exists in C since V is complete and C isclosed. We have then

.g ( f. = .g ( lim n %!fn. = limn+!

.g ( fn. = d .

The uniqueness of f can be easily obtained by contradiction. 34



This theorem gives an existence and uniqueness result which is often used inoptimization problems. However, it is of limited practical usefulness for findingthis optimal point. The following theorem provides a useful characterization.

Corollary 1.3.2. Let C be a closed convex subset of a real Hilbert space V .For f # C and g # V , the following assertions are equivalent

1. .g ( f. = infh#C .g ( h.2. (g ( f, h( f) ' 0, for all h # C.

Theorem 1.3.6 (Projection theorem). Let S be a closed subspace of aHilbert space V . Then every g # V can be uniquely decomposed as g = f + hwhere f # S and h # S), and thus we write symbolically

H = S 2 S) .

The space V is the direct sum of S and its orthogonal complement S). Inother words, to every g # V , we can assign a unique element f which is theelement in S closest to f . This property allows us to consider the operatorPsg = f called the orthogonal projection corresponding to S. Note that wehave

P 2S = PS , and (PSf, g) = (f, PSg) , for every f, g # V .

Clearly, we have also PS#g = g ( PSg = h, with h # S).

1.3.3 Bounded linear operators on Hilbert spaces

Linear operators on normed and Hilbert spaces play an important role inapplied mathematics. We turn now to linear functionals A : V % C and wewill consider also bilinear functionals and quadratic forms.

We recall that a linear operator A : X % Y , where X, Y are normedlinear spaces, is called bounded if the operator norm is finite, i.e., if thereexists M > 0 such that

.Af.Y ' M.f.X , for all f # X .

A consequence of the linearity of the operator is that continuity can be checkedat a single point only.

Lemma 1.3.6. Consider a linear operator A : X % Y , where X, Y arenormed linear spaces. The operator A is continuous at every point if it iscontinuous at a single point.

And according to Lemma 1.1.3, a linear operator A : X % Y between twonormed linear spaces is continuous if and only if it is bounded on X.

One of the most important operator for our purposes is obviously thedi!erential operator, defined on the space of di!erentiable functions on aninterval [a, b] " R, by



(Df)(x) =df(x)dx

= f '(x) .

However this operator is unbounded.The Cauchy-Schwarz inequality shows that the linear functional A : f 7%

(g, f) is bounded for the norm .g.. Interestingly, in a Hilbert space, everybounded linear functional can be written in this way.

Theorem 1.3.7 (Riesz lemma). Let A be a bounded linear functional on aHilbert space V . Then, there is a unique g # H such that Af = (g, f) for allf in V . Moreover, we have .A. = .g..

Proof. If A 8 0 then we set g = 0. Otherwise, consider the kernel of A,Ker(A) = {f # V , Af = 0}. Since A is linear and bounded, Ker(A) is aclosed proper subspace of V , and let g be a unit vector of Ker(A)). For everyf # V , we write Afg (Agf # Ker(A) and thus we have

0 = (g, Afg (Agf) = Af (Ag (g, f) .

Consequently, we set g = (Ag)g. The uniqueness is obtained by consideringg1, g2 two such vectors. Then, for any f # V ,

(g1 ( g2, f) = (g1, f)( (g2, f) = Af (Af = 0

which means that g1(g2 # V ) = {0}. Finally, the equality .A. = .g. resultsfrom the definition of .A. and the Cauchy-Schwarz inequality. 34

This result indicates that a Hilbert space is equivalent to its dual space, H =H '. In other words, any continuous linear functional given on a Hilbert spacecan be uniquely identified with an element in the same space, i.e., H and H '

are isomorphic.We have two results involving bilinear functionals defined on a Hilbert

space.

Lemma 1.3.7. Let A be a bounded operator on a Hilbert space V . Then, thebilinear functional a(·, ·) ( i.e., linear in each variable) defined by a(u, v) =(Au, v) is bounded and .A. = .a..

Lemma 1.3.8. Let a(·, ·) be a bounded bilinear functional on a Hilbert spaceV . There exists a unique bounded operator A on V such that

a(f, g) = (Ag, g) for all f, g # V .

A bilinear functional a(·, ·) on a normed space V is called elliptic or coerciveif there exists a constant % > 0 such that

a(f, f) & .f.2 , for all f # V .

The following theorem is a generalization of the Riesz lemma, it has importantapplications to boundary value problems (cf. Chapter 2).



Theorem 1.3.8 (Lax-Milgram theorem). Let a(·, ·) be a bounded ellipticbilinear functional on a Hilbert space V . For every bounded linear functionalA on V there exists a unique u # H such that

a(f, g) = Ag , for all g # V .

Proof. Lemma 1.3.8 states that there exists a bounded operator A such that

a(f, g) = (Af, g) , for all f, g,# V .

Since the bilinear functional a is elliptic, we have then

%.f.2 ' a(f, f) = (Af, f) ' .Af..f. ,

and then we write

.Af. & %.f. , for all f # V .

This indicates that A is bounded (or continuous). Moreover, A is a one-to-onemap. Indeed, consider f1, f2 # V . If Af1 = Af2 then A(f1 ( f2) = 0 and thus

.f1 ( f2. '1%.A(f1 ( f2). = 0 .

It remains to show that the range of A is V . Let (fn)n"1 be a sequence in V .If .Afn ( g. tends to 0 for some g # V then

.fn ( fm. '1%.Afn (Afm. % 0 , as m, n %! .

Hence, (fn)n"1 is a Cauchy sequence. The space V is complete, there existsf # V such that .fn(f. % 0 and thus .Afn(Af. % 0 since A is continuous.We deduce that Af = g and g # Im(A). The range of A is closed. SupposeIm(A) )= V , then there is an element g )= 0 # H such that g 1 Im(A), i.e.,

(Af, g) = 0 , for all f # V .

In particular, we have

0 = |(Af, f)| = |a(f, f)| & %.f.2 ,

which contradicts the assumption g )= 0 and completes the proof. 34

The Lax-Milgram theorem is especially used to establish the existence anduniqueness of weak solutions of boundary-value problems as will be seen inthe next Chapter.

A bounded linear operator A on a Hilbert space V is said to be invertible ifthere exists a bounded operator A*1 such that AA*1 = A*1A = idV and A*1

is then called the inverse of A. In finite dimensional spaces, if the operatoris injective and an isometry it is invertible. Notice that A may have a linearspace inverse on V without being invertible. The following result provides acriterion for invertibilty.



Lemma 1.3.9. Let A be a bounded linear operator on a Hilbert space V , with.A. < 1. Then the operator I (A is invertible and its inverse is given by

(I (A)*1 =!$

n=0

An .

Corollary 1.3.3. Consider two linear operators A, T defined on a Hilbertspace V and suppose that T is invertible and that .T (A. < .T*1.*1. ThenA is invertible.

By analogy with linear algebra (see Chapter 4), we will define the spectrumof a linear operator A in the set L(V ) as the set of complex numbers ( suchthat A ( (I is not invertible in V , where I is the identity operator. And weobserve that the spectrum of an operator on a finite-dimensional vector spaceis precisely the set of eigenvalues. It can be shown that the spectrum of everyelement A in L(V ) is a compact and nonempty set and is contained in thedisk {( # C , |(| '. A.}.

1.3.4 Compact linear operators

An important class of operators is the class of compact operators. A linearoperator A defined on a normed space V is called compact if for every sequence(fn)n"1, the sequence (Afn)n"1 has a convergent subsequence in V wheneverfn is bounded. The set of all compact operators is often denoted by C(V ).

Lemma 1.3.10. Every compact linear operator is bounded. A linear combi-nation of compact operators is bounded.

Lemma 1.3.11. Let V be a Hilbert space and let (An)n"1 be a convergentsequence of compact operators. Then the limit A is a compact operator.

One of the most important example of compact operator is the integral oper-ator, defined by

(Kf)(x) =! b

aK(x, y)f(y)dy .

The function K is called the kernel of the operator. For instance, if! b

a

! b

a|K(x, y)|2dydx < !

then K is a bounded operator on L2([a, b]) and

.K. '-! b

a

! b

a|K(x, y)|2dydx

.1/2

.



Theorem 1.3.9 (Arzela-Ascoli). Consider a uniformly equicontinuous se-quence of functions (fn)n"1 on a compact interval, i.e., such that for every& > 0 there exists ' > 0 such that

|fn(x)( fn(y)| ' & if |x( y| < '.

If the sequence if (fn)n"1 bounded, then it admits a uniformly convergentsubsequence.

A linear operator A defined on an inner product space V is called sym-metric if its domain is dense and if, for every f, g in V

(g,Af) = (Ag, f) .

Bounded symmetric operators are also called Hermitian. A complex numberz is called eigenvalue of a symmetric linear operator A if there is a nonzerovector u # V such that Au = zu. The vector u is then called a correspondingeigenvector.

Lemma 1.3.12. Let A be a symmetric linear operator. Then all eigenvaluesare real and all corresponding eigenvectors are orthogonal.

Theorem 1.3.10. Any symmetric compact operator A has an eigenvalue (1

such that |(1| = .A..

Proof. We pose ( = .A., with ( )= 0 and we observe that

.A.2 = sup&f&=1

.Af.2 = sup&f&=1

(Af, f) = sup&f&=1

(f, A2f) ,

then there exists a sequence fn such that

limn+!

(fn, A2fn) = (2 .

The operator A being compact, we assume that limn+!A2fn = (2f , i.e.,A2fn is convergent and thus we write

.(A2 ( (2)fn.2 = .A2fn.2 ( 2(2(fn, A2fn) + (4

' 2(2((2 ( (fn, A2fn))

and thus limn+!(A2fn ( (2fn) = 0. And we deduce that limn+! fn = f .We notice that u is a normalized eigenvector of A2 since (A2 ( (2)u = 0.This equation can be factorized as (A + ()(A ( ()u = 0 which is equivalentto (A( ()u = v and (A + ()v = 0. Hence, v is an eigenvector correspondingto (( or v = 0. This shows that u )= 0 is an eigenvector of (. 34

If the operator A is bounded, the largest eigenvalue in modulus cannot belarger than .A..

Compact operators are very similar to matrices, in the sense that thespectrum of a compact operator is similar to the spectrum of a finite matrix.



Theorem 1.3.11. Let A be a compact symmetric operator on a Hilbert spaceV . Then the spectrum of A is either a finite set or a sequence of real eigenval-ues converging to 0. In this case, the corresponding normalized eigenvectorsuj form an orthonormal set and every f # V can be decomposed as

f =!$

k=1

(uk, f)uk + g ,

where g # Ker(A).

1.3.5 Adjoint and self-adjoint operators

The spectral theory of continuous self-adjoint linear operators on a Hilbertspace generalizes the usual spectral decomposition of a matrix. We introducea conjugation for operators on Hilbert spaces.

Let A be a bounded linear operator on a Hilbert space V . The operatorA( : V % V defined by

(Af, g) = (f, A(g) , for all f, g # V ,

is called the adjoint operator of A.

Proposition 1.3.1. The adjoint operator satisfies the following properties

1. (A + B)( = A( + B(;2. ((A)( = (A(;3. (AB)( = B(A(;4. (A()( = A;

Lemma 1.3.13. The adjoint operator A( of a bounded operator A is bounded.Furthermore, the following identities hold

.A. = .A(. , and .A(A. = .AA(. = .A.2 .

A special case is when operators A and A( coincide. If A = A(, i.e., if we have(Af, g) = (f, Ag), for all f, g # V , then the operator A is called self-adjoint orHermitian. If A = (A(, the operator A is called anti-Hermitian. For instance,let A be the operator on L2([a, b]) defined by

(Af)(x) = xf(x) , for all x .

We have then

(Af, g) =! b

axf(x)g(x) dx =

! b

af(t)xg(x) dx = (f, Ag) ,

and thus A is self-adjoint.Let a(·, ·) be a bounded bilinear functional on a Hilbert space V and let

A be an operator on V such that a(f, g) = (Af, g) for all f, g # V . Then A isself-adjoint if and only if a(·, ·) is symmetric.



Lemma 1.3.14. Let A be a bounded operator on a Hilbert space V . Then theoperators A(A and A + A( are self-adjoint. Furthermore, the product of twoself-adjoint operators is self-adjoint if and only if the operators commute.

We consider the projection operator PS (cf. Section 1.3.2) which maps everyelement of H onto its orthogonal projection in the subspace S.

Lemma 1.3.15. For every subspace S of a Hilbert space V , the projectionoperator PS is a Hermitian (self-adjoint) operator and is an idempotent op-erator, i.e., such that P 2

S = PS. Conversely, any idempotent operator is aprojection operator onto a given subspace.

The following property of Hermitian operators is related to the spectral prop-erties of these operators.

Proposition 1.3.2. If A is a Hermitian operator in a Hilbert space V , then

.A. = sup&f&=1

|(Af, f)| .

A bounded operator A on a Hilbert space V is called isometric if .Af. = .f.for every f # H. It is called unitary if A(A = AA( = I on the Hilbert spaceV . Every unitary operator is isometric, i.e., it preserves the distance betweenany two elements in V . Furthermore, if A is unitary so are A*1 and A(. Forexample, consider the operator A on L2([0, 1]) defined by

(Af)(x) = f(1( x) .

It is easy to show that A = A( = A*1 and thus A is unitary.

Lemma 1.3.16. A bounded operator A on a Hilbert space V is isometric ifand only if A(A = I on V .

An operator A defined on a Hilbert space V is called positive if it is Hermitianand if (Af, f) & 0, for all f # V . Consider for instance the integral operatorK on L2([a, b]) defined by

(Kf)(x) =! b

aK(x, y)f(y)dy

where K(·, ·) is a positive continuous functions defined on [a, b]. Then, K is apositive operator since we have, for all f # L2([a, b])

(Kf, f) =! b

a

! b

aK(x, y)f(y)f(y) dydx =

! b

a

! b

aK(x, y)|f(y)|2 dydx & 0 .

Lemma 1.3.17. Let A be a bounded operator on a Hilbert space V . Then theoperators A(A and AA( are positive. Furthermore, if A is invertible then A*1

is positive.



A square root of a positive operator A on a Hilbert space V is a self-adjointoperator B such that B2 = A. It can be shown that every positive operatorhas a unique positive square root.

We define the resolvent operator of an operator A on a Hilbert space V asthe operator A$ = (A( (I)*1. As for bounded operators, a complex number( is called a regular value for A if the domain of the resolvent operator isthe whole space V . The set of complex numbers ( which are not regularis called the spectrum of A. The set of all eigenvectors corresponding to aneigenvalue ( is called the eigenspace of ( and di!erent spaces correspondingto distinct eigenvalues of a Hermitian or unitary operator are orthogonal toeach other. All eigenvalues of a Hermitian operator are real. Furthermore,all eigenvalues of a positive (resp strictly positive) operator are nonnegative(resp. positive). All eigenvalues of a unitary operator on a Hilbert space Vare complex numbers of unit module.

Lemma 1.3.18. Let P be an invertible operator on a Hilbert space V . Then,for any operator A defined on V , the operators A and PAP*1 have the sameeigenvalues.

Proposition 1.3.3. If A is a bounded Hermitian operator defined on a Hilbertspace V then the eigenvalues of A are such that

|(| ' sup&f&=1

(Af, f) .

Furthermore, if A is a compact Hermitian operator on V , then there is anelement g such that .g. = 1 and

|(Ag, g)| = sup&f&=1

|(Af, f)| .

Theorem 1.3.12 (Spectral theorem). Let V be an infinite dimensionalHilbert space and A be a nonzero, compact and Hermitian operator on V .Then there exists a sequence of nonzero real-valued eigenvalues ((n)n"1 and acorresponding sequence of eigenvectors (un)n"1 of A forming an orthonormalbasis. Moreover, for each f in V , we have

Af =!$

k=1

(k(f, uk)uk .

Furthermore, let Pn be the projection operator onto the space spanned by(un)n"1. Then, for all f in V we have

f =!$

n=1

Pnf , and A =!$

n=1

(nPn .



1.3.6 The Fourier transform

In Section 1.2, we introduced Lebesgue Lp spaces and related main properties.The space L2 of square integrable functions is undoubtedly the most importantLp-space. It has a key role in Fourier analysis5. Here, we start by consideringthe Fourier transform in L2(R) from the historical point of view of Fourier.Given an arbitrary function f defined on [(", "], if f can be decomposed as

f(x) =a0

2+

!$

k=1

ak cos(kx) + bk sin(kx)

then the coe"cients must be given by, for all k & 1

ak =1"

! #

*#f(x) cos(kx) dx and bk =

1"

! #

*#f(x) sin(kx) dx .

Rapidly, the questions of the existence of such decomposition, of the con-vergence of the series, and the type of convergence, of the limit arise. Weshow now that all these questions can be addressed in the context of an innerproduct space or a Hilbert space.

Given f # L1(R), its Fourier transform is the function f defined by

f(k) =162"

! !

*!f(x) exp((ikx) dx (1.9)

where dx denotes the ordinary Lebesgue measure. It may not be obvious thatthis definition is justifiable for all functions in L1(R), although the integralis well-defined for all real k. Nevertheless, we shall notice that the functionexp((ikx) is continuous and bounded and thus the product f(x) exp(ikx) islocally integrable for any k # R. Furthermore, we have

|f(x) exp((ikx)| ' |f(x)| ,

and thus the integral (1.9) is well defined for all k # R.For example, consider the function f(x) = exp((%|x|) depicted in Fig-

ure 1.1 (left-hand side), for % = 1. We have then

f =162"

! !

*!exp((%|x|) exp((ikx) dx

=162"

"! 0

*!exp((%( ik)x) dx +

! !

0exp(((% + ik)x) dx

#

=162"

"1

%( ik+

1% + ik

#

=162"

2%

%2 + k2=

62%6

"(%2 + k2),

5 Named after the French mathematician Joseph Fourier (1768-1830) known forinvestigating the decomposition of general functions as infinite sums of simplerfunctions, the Fourier series, and their application to problems of heat flow.



0

0.2

0.4

0.6

0.8

1

-1.5 -1 -0.5 0 0.5 1 1.5 0

0.2

0.4

0.6

0.8

-15 -10 -5 0 5 10 15

Fig. 1.1. A function f ! L1(R) (left) and its Fourier transform (right).

and the Fourier transform f is represented in Figure 1.1 (right-hand side). Thefollowing properties come almost directly from the definition of the Fouriertransform.

Lemma 1.3.19. Given f, g # L1(R) and % # C, we have

1. .f.! ' .f.1, where .f.1 =! !

*!|f(x)| dx;

2. f is uniformly continuous;3. !f + g = f + g and 8%f = %f .

The Fourier transform of integrable functions has additional properties.

Lemma 1.3.20. Let (fn)n"1 be a sequence in L1(R) such that .fn. % 0 whenn %!. Then, the sequence (fn)n"1 converges uniformly to f on R.

Theorem 1.3.13 (Riemann-Lebesgue). If f # L1(R) then |f(k)| % 0 as|k| %! and thus f # Cc(R), where Cc(R) denotes the space of continuousfunctions on R vanishing at infinity ( i.e., compactly supported functions).

Proof. The proof is found in most classical analysis books. We remind thatexp((ikx) = exp((ikx( ") and thus by substitution, we write

f(k) =12

162"

"! !

*!f(x) dx(

! !

*!f(x( "/k) exp((ikx) dx

#

=12

162"

! !

*!(f(x)( f(x( "/k)) exp((ikx) dx .

And we deduce then

|f(k)| ' 12

162"

! !

*!|f(x)( f(x( "/k))| dx .



Theorem 1.2.2 allows to conclude that lim|k|+!

|f(k)| = 0. 34

This theorem is notably used to prove the validity of asymptotic approxima-tions for integrals. From these results, we observe that the Fourier transform isa continuous linear operator from L1(R) to the normed space (Cc(R), . ·.!).

Convolution theorem

Before going further on the analysis of the Fourier transform for functionsin L2(R), we need to introduce the convolution of two functions. If f, g arefunctions in L1(R), their convolution is the product f 9 g defined by

(f 9 g)(x) =162"

! !

*!f(x( y)g(y) dy .

Proposition 1.3.4. Convolution satisfies the same algebraic properties asclassical multiplication, i.e., for f, g # L1(R)

1. f 9 (%g + *h) = %(f 9 g) + *(f 9 h), for any %,*;2. f 9 g = g 9 f ;3. f 9 (g 9 h) = (f 9 g) 9 h.

And we have the following result.

Theorem 1.3.14 (Convolution theorem). Given f, g # L1(R), we have!f 9 g = f g.

Corollary 1.3.4. Suppose f, g # L1(R) and % is a real number. Then

1. if g(x) = f(x) exp(i%x) then g(k) = f(k ( %);2. if g(x) = f(x( %) then g(k) = f(k) exp((i%k);3. if h = f 9 g then h(k) = f(k)g(k);4. if g(x) = f((x) then g(k) = f(k);5. if g(x) = f(x/%) for % > 0 then g(k) = %f(%k);6. if g(x) = (ixf(x) then f is di!erentiable and f '(k) = g(k).

This shows that the Fourier transform converts the convolution to a pointwiseproduct as it converts the multiplication by a character to a translation. Werecall that a function + is a character of R if |+(x)| = 1 and if, for all x, y # R,+(x + y) = +(x)+(y).

We can also observe that the Fourier transform of (f(x + %) ( f(x))/%is f(k)(exp(i%k)( 1)/%. And this suggest that the Fourier transform of f ' ifikf(k), if f, f ' # L1 and if f is the integral of f '. This interesting feature isextremely useful in the analysis of di!erential equations.



Inversion theorem

It is surely interesting to return from the Fourier transforms to the functions,i.e., to have an inversion formula. Hence, if f, f # L1(R) we would expect aformula like

f(x) =162"

! !

*!f(k) exp(ixk) dk . (1.10)

If f # L1(R), the righ-hand side is well defined but a proof of this identity(which is true indeed) is di"cult and not straightforward. However, when bothf and f are integrable, the following inverse equality holds for almost every xNotice that even if f # C0(R) the integral 1.10 will not converge for generalf # L1(R). However, this integral converges absolutely for a dense subspaceas will be seen later. Actually, we have the following results.

Theorem 1.3.15 (Inversion theorem). If f, f # L1(R) and if

g(x) =162"

! !

*!f(k) exp(ixk) dk , x # R ,

then g # Cc(R) and f(x) = g(x) a.e.

Corollary 1.3.5 (Uniqueness theorem). If f # L1(R) and f(k) = 0 forall k # R then f(x) = 0 a.e.

Plancherel theorem

Since the Lebesgue measure of R is infinite, the set L2(R) of square integrablefunctions is not a subset of L1(R). Hence, we cannot extend the definition ofthe Fourier transform (1.9) directly to every function f # L2(R). However, iff # L1 * L2 then f # L2 and the following Plancherel identity, holds

.f.2 = .f.2 .

The isometric mapping of L1(R) * L2(R) into L2(R) has a unique extensionto a linear mapping from L2(R) to L2(R) and this extension will be called theFourier transform of every f # L2(R). We have the following theorem.

Theorem 1.3.16 (Plancherel theorem). For every f # L2(R) there existsa function f # L2(R) such that

1. if f # L1(R) * L2(R) then f(k) = 1,2#

0!*! f(x) exp((ikx) dx is the

Fourier transform of f ;2. for every f # L2(R), we have .f.2 = .f.2 (Plancherel identity);3. the mapping f % hatf is a Hilbet space isomorphism of L2(R) onto L2(R);



4. the following symmetric relations exist between f and f

.f ( +.2 % 0 as n %!, where +(k) =162"

! n

*nf(x) exp(ikx) dx and

.f ( ,.2 % 0 as n %!, where ,(x) =162"

! n

*nf(k) exp(ixk) dk.

This result indicates that the Fourier transform of any L2(R) function f isanother square integrable function. The first two properties determine themapping f % f uniquely, since L1 * L2 is dense in L2. The fourth propertyis called the inversion theorem in L2(R).

Theorem 1.3.17 (Parseval identity). If f, g # L2(R) then we have! !

*!f(x)g(x) dx =

! !

*!f(k)g(k) dk .

The Fourier transform can be extended to functions in L1(Rn) as follows

f(k) =1

(2")n/2

!

Rn

f(x) exp((i(k, x)) dx ,

where x and k are n-dimensional vectors and (k, x) denotes here the usualdot product of the vectors, i.e., (k, x) =

'nj=1 kjxj . All the basic properties

listed above still hold.

1.4 Distributions

In this section, we shall provide a condensed introduction of the main conceptsand properties of distribution theory which will be required in the next sectionand the following chapters. The reader is referred to the book of L. Schwartz[Sch50] for a detailled analysis of the topics introduced here. The current formof distribution theory, which is largely based on Fourier transformation, is dueto L. Schwartz6, is now an essential part in the analysis of partial di!erentialequations.

From Section 1.2, we know that the Lebesgue Lp spaces contain functionsnon regular and non continuous functions for which derivatives are not de-fined in the classial sense. Nonetheless, the classical derivatives exist almosteverywhere. Thus, it seems reasonable to generalize the notion of derivativeto be independent of zero-measure sets. This is the purpose of the conceptof weak derivative introduced by J. Leray (1906-1998) and S.L. Sobolev. Anexample of singular function is the delta function ', introduced by P. Dirac(1902-1984), characterized by the following identity6 French mathematician (1915-2002) who introduced distributions as objects which

generalize functions. The theory of generalized function was initiated in 1935 bythe Russian mathematician S.L. Sobolev (1908-1989).


1.4 Distributions 43

! !

*!f(x)'(x) dx = f(0) (1.11)

for any continuous function f . No classical function satisfies this prop-erty. Indeed, from this definition, it results that '(x) = 0 for x )= 0 and0!*! '(x) dx = 1, for f(x) = 1. Hence, the intuition would guide us to see the

delta function as being the limit of a sequence of ordinary functions fn suchas Gaussian functions

limn+!

! !

*!f(x)fn(x) dx = f(0) , with fn(x) =

n62"

exp"( (nx)2

2

#.

The purpose of the distribution theory is to provide a convenient setting fordefining delta-like functions as continuous functionals in a space of regulartest functions.

1.4.1 Test functions

We remind that the support of a function f : Rn % R, denoted by supp(f),is the closure of all points x such that f(x) )= 0. A function f on Rn hasa compact suppport or is compactly supported if its support is a compact(i.e., closed bounded) subset of Rn. The space of all functions f # Ck(Rn)having a compact support is denoted Ck

c (Rn) or sometimes Dk(Rn). A testfunction is an infinitely di!erentiable function on Rn with compact support.The space C!c (Rn) of all test functions is denoted by D(Rn) or simply by D.It results from the definition that a test function f is a infinitely di!erentiablefunction vanishing outside a bounded set. Moreover, the space D(Rn) is densein L1(Rn).

In solving boundary value problems, distributions on an open subset $of Rn are needed. The concept of test function can be extended to open setsquite naturally. We consider the space D($) of test functions on $ as thespace of all functions f # C!($) with support contained in a compact subsetof $.

The existence of a test function is not obvious. A classical example is givenby the function +(x) = f(|x|2 ( 1) where |x|2 = x2

1 + · · · + x2n and

f(t) = exp(1/t) , if t < 0 and f(t) = 0 , if t & 0 .

By a suitable translation and scaling, we show that

+ # C!C (Rn) ,

!+(x) dx = 1 , + & 0 , supp(+) = {x; |x| ' 1} .

Using this example, we can obtain new test functions by taking an integrablefunction u and defining the convolution

u%(x) =!

u(x( &y)+(y) dy = &*n

!u(y)+((x( y)/&) dy .



Other examples of test functions are +(%x + *), %,* # R, +(k)(x) for k > 0,and f(x)+(x) where f is an arbitrary infinitely di!erentiable function. Theexistence of test functions allows t prove the following result.

Proposition 1.4.1. If f, g are continuous functions in Rn and if the followingidentity holds

! !

*!f(x)+(x) dx =

! !

*!g(x)+(x) dx , for every + # C!c (Rn) ,

then f = g.

This result explains the name of test function. Instead of testing the equal-ity at any point x # Rn, the functions f and g are tested using infinitelydi!erentiable functions with compact support.

A sequence of test functions (+n)n"1 # D(Rn) is said to converges to orderm to a function + # D(Rn) if

1. the function +n and + all have supports within a common bounded setK " Rn;

2. Dk+n % Dk+ uniformly for all x for all orders k = 0, . . . ,m, where Dk

denotes the partial derivatives of order k (cf. Equation (1.2)).

If the convergence to order m is achieved for all m & 0 then (+n)n"1 is saidto converge to +. This definition still holds when replacing Rn by an open set$ " Rn.

1.4.2 Distributions

The main idea is to identify arbitrary functions with linear functionals in thespace of test functions. A distribution of order m is a linear functional T onD(Rn) which is continuous to order m, i.e., such that

1. T (a+ + b,) = aT (+) + bT (,), for every a, b # C and +, , # D(Rn),2. T (+n) % T (+) whenever +n % + is any sequence of functions in D(Rn)

convergent to order m.

A linear function T that is continuous with respect to sequences (+n)n"1 thatare convergent to all orders is simply called a distribution on Rn. The spaceof all distributions is the dual space of D(Rn) and is denoted by D'(Rn) orsimply D'.

The space L1 being a Hilbert space, we have seen that this space and itsdual can be identified through the inner product (·, ·), thanks to the Rieszrepresentation theorem 1.3.7

u : L1(Rn)$ L1(Rn) % R , (u, +) 7% /u, +0 = (u, +)L1(Rn) =! !

*!u+ .



The bilinear form (u, +) 7% /u, +0 is continuous and still makes sense if thefunctions + are taken in a subspace V of L1. Hence, every locally integrablefunctions f on Rn can be identified with a distribution T as

/T, +0 = (f, +) =! !

*!f+ .

Formally we introduce the following definition.

Definition 1.4.1. A distribution T in Rn is a linear form on D(Rn) suchthat to every compact set K " Rn, there exist constants C and m such that

|/T, +0| = |T (+)| ' C$

|k|%m

sup |Dk+| , + # C!c (K) .

Conventionally, a function f # Lp(Rn) will be identified to its associateddistribution T # D'(Rn) and the pairing between f and a test function + iswritten as

(f, +) = /T, +0 .

A distribution T # D' is called regular if there exists a locally integrablefunction f such that

/T, +0 =! !

*!f+ , for every + # D .

The distribution is called singular if it is not regular.

We provide an important example of such distribution. Given a # R wedefine the distribution 'a on D(R) by

/'a, +0 = 'a(+) = +(a) , for every + # D(R) .

The map 'a : D(R) % R is linear and is continuous. Moreover, we have

|/'a, +0| ' .+.L"(R) , for every + # D(R) .

Hence, 'a is a distribution, called the Dirac mass at point a. But, it cannotbe represented by any locally integrable function (cf. the introduction of thischapter). The Dirac delta is thus a singular distribution.

An equivalent form of the definition 1.4.1 is given by the following theorem.

Theorem 1.4.1. A linear form T on D(Rn) is a distribution if and only ifT (+n) % 0 when n %! for every sequence (+n)n"1 in D(Rn) such that

1. D"+n % 0 uniformly when n %!, for every multiindex %;2. there is a fixed compact subset of Rn containing the support of all +n.

A sequence satisfying these two conditions is said to converge to 0 in D(Rn).



1.4.3 Operations on distributions

We define now several usual operations on distributions that are already fa-miliar for classical infinitely di!erentiable functions.

If T and S are distributions of order m on Rn, it is easy to see that T + Sand %T are distributions of order m, for all % # R. This shows that D'm(Rn) isa vector space. However, it has been proved by Schwartz that an associativemultiplication of two distributions cannot be defined, i.e., the product ofarbitrary distributions is not a distribution. For example, suppose we define/ST, u0 = /S, u0/T, u0, this would clearly not be linear in u. However, if , is afunction in Cm(Rn) and T a distribution of order m then ,T can be definedas a distribution of order m, simply by posing

/,T, +0 = /T, ,+0

since ,+ is in Cmc (Rn) for every + # Cm

c (Rn). And we observe that thisdefinition still holds if , is not compactly supported.

Suppose $ is an open subset in Rn and consider a distribution T on Rn.We define the restriction of T to the open set $, denoted by T|! , as

/T|! , +0 = /T, +0 , for every + # C!c ($) .

Furthermore, if f s a locally integrable function we have:!

!f|!(x)+(x) dx =

! !

*!f|!(x)+(x) dx , for every + # C!c ($) .

And we observe that this definition coincide with the notion of restriction forclassical functions. However, the restriction of a distribution is only definedfor an open set.

Di!erentiation

One of the strongest advantage of the distribution theory is that the di!eren-tiation operation is always defined and every distribution has all derivativesthat are distributions. Let consider a di!erentiable function f # C1(Rn) and+ # D1(Rn), then using integration by parts

9!f

!xi, +

:=

! !

*!

!f

!xi(x)+(x) dx ,

= [+f ]!*! (! !

*!

!+

!xi(x)f(x) dx = (

9f,

!+

!xi

:.

This result is still valid if we set f to be a distribution. More precisely, letT # D'(Rn) be a distribution on Rn. Then the derivative !T/!xk is definedby



9!T

!xk, +

:= (

9T,

!+

!xk

:.

More generally, given a multiindex %, then the linear functional denoted byD"T is defined by

/D"T, +0 = /T, ((1)|"|D"+0 = ((1)|"|/T, D"+0 .

Proposition 1.4.2. If T is a distribution on Rn then D"T is a distributionon Rn for any multiindex %.

Proof. Linearity is obvious to show. To prove that D"T is continuous, weuse the definition of convergence of a sequence of functions (+n)n"1 to + inD which requires that all derivatives up to and including orde % convergeuniformly on a compact subset. Hence, D"+n tends to D"+ in D and thenD"T is a continuous fonctional and thus D"T is a distribution. 34

Set H to be the Heaviside function of a single variable x (i.e., the char-acteristic function of R+), H(x) = 1 if x & 0, and H(x) = 0 if x < 0. Itis clear that H is not di!erentable in the usual sense, but it is obviously alocally integrable function and it generates a regular distribution. Indeed, forany test function + # D(R), we have

9!H

!x, +

:=

9dH

dx, u

:= (

! +!

*!+'(x)H(x) dx

= (! +!

0+'(x) dx = +(0) = /'x=0, +0 .

Hence, the derivative of the Heaviside function H in the sense of distributionsis the Dirac mass and we will write as for functions '0 = H '.

A sequence of regular distributions (Tn)n"1 (generated by a sequence offunctions (fn)n"1) is said to converge weakly to a distribution T if

/Tn, +0 % /T, +0 , for every + # D .

This notion is called the weak distributional convergence. For example, thesequence of functions on R defined by

fn(x) =n

"(1 + (nx)2), n & 1 ,

is distributionally convergent to the Dirac delta distribution ' (cf. Figure 1.2).

Lemma 1.4.1. Let Tn % T in D'(R). Then, for every multiindex %

D"Tn % D"T .



0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

-3 -2 -1 0 1 2 3

f1f2f3f5

Fig. 1.2. Convergence of a sequence of distributions to the Dirac delta function !.

Let T be a distribution in D'(R). The antiderivative (primitive) of T isa distribution S on R such that S' = T . For every distribution T defined onR, there exists an antiderivative distribution S defined to within a constantterm. Indeed, consider a test function +0 # D(R) such that

! !

*!+0(x) dx = 1 .

Then, every function + # D(R) can be uniquely decomposed as

+ = C+0 + +1 , with C =! !

*!+(x) dx , and

! !

*!+1(x) dx = 0 .

Given F # D'(R) we define a functional T on D by

/S, +0 = /S, C+0 + +10 = CK ( /T, ,0

where K is a constant and , is the test function

,(x) =! x

*!+1(x) dx .

Then S' = T and S is a distribution. We have then the following result.

Lemma 1.4.2. Given T # D'(R). The distributions for which T ' = 0 are theconstant functions.

Lemma 1.4.3. Consider $ =]0, 1[. Then, for every funtion f # L1(R), wehave

d

dx

"! x

0f

#= f , in D'($) .



1.4.4 Fourier transform

In Section 1.3.6 we have recalled the notion of Fourier transform applicableto any function f such that |f | is integrable over R. The Fourier transform ofa distribution T can be defined as the distribution T given by

F(/T, +0) = /T, +0 ,

for all test function + (and we used the convenient notation F(·) for theFourier transform of an expression). If the inverse Fourier transform is definedon distributions by F*1(/T, +0) = /T, +*10 then

F*1F/T, +0 = /T, +0 .

However, the Fourier theorem requires the function + to be of bounded varia-tion for the result to hold. If + is a function of bounded support, the function+ cannot be of bounded support in general. Hence the term /T, +0 may notbe well-defined. We need to introduce another function space to define theFourier transform of distributions.

A function + is called rapidly decreasing as |x|%! if the equality

lim|x|+!

|x|m+(x) = 0

holds for an arbitrary m. Such a function approches 0 as |x|%! faster thanany inverse power |x|*n. The space of test functions S(R) consists of functionsrapidly decreasing as |x|%! together with derivatives of every order up toorder p, i.e.,

S(R) = {+ ; suppx#R |xm+(p)(x)| < ! , for all m, p # N(} .

S(R) is called the space of rapidly decreasing functions. A sequence of func-tions (+n)n"1 of S(R) is said to converge to + if the equality

limn+!

supx#R

|xm(+(p)n (x)( +(p)(x))| = 0 , 0 ' |p| ' m ,

is satisfied for an arbitrary m. The space of linear functions on S(R) is denotedS '(R) and these functions are called tempered distributions. Since every testfunction is a rapidly decreasing function then D(R) " S(R). If T is a tem-pered distribution, then the Fourier transform is well-defined, i.e., the Fouriertransform of a rapidly decreasing function is another rapidly decreasing func-tion.

For a locally integrable function f on R identified with a distribution T ,we have

F(/T, +0) = F(f, +) =162"

! !

*!

"! !

*!exp((ikx)+(x) dx

#f(k)dk

=162"

! !

*!+(x)

"! !

*!exp((ikx)f(k) dk

#dx

=! !

*!+(x)f(x) dx = /T, +0 .



With this definition, the Fourier transform of the Dirac distribution can bedefined by

F(/'a, +0) = /'a, +0 = +(a) =162"

! !

*!exp((iax)+(x) dx .

1.5 Sobolev spaces

In this section, we introduce Sobolev spaces of integer order and we providesome of their basic properties. Our intent is to deal later with partial dif-ferential equations and linear operators and we will see that Sobolev spacesare designed to find the correct spaces for solutions. In this section, we areconsidering functions for which all the derivatives, in the distributional sense,belong to L2 space and such that most of the classical derivation rules canbe applied. Before turning to Sobolev spaces, we briefly recall the simplerconcept of Holder spaces.

1.5.1 Holder spaces

We remind that there exists a condition for checking the regularity of real-valued functions, stronger than the regular continuity, i.e., the Lipschitz con-tinuity7. This condition is central to the Cauchy-Lipschitz theorem (or thePicard-Lindelof theorem) which states and ensures the existence and unique-ness of the solution to an initial value di!erential problem. More specifically,a continuous real-valued function f defined on an open subset $ of Rn is saidto be Lipschitz continuous if there exists a nonnegative real constant k suchthat the following estimate holds

|f(x)( f(y)| ' k .x( y. , for all x, y # $ .

The function f is called locally Lipschitz continuous if for every x # $ thereexists a neighborhood U of x such that f |U is Lipschitz continuous. A real-valued function f is said to be Holder continuous if there exist nonnegativereal constants k and m such that

|f(x)( f(y)| ' k .x( y.m , for all x, y # $ .

The number m is called the Holder exponent and we observe that if m = 1,then the function satisfies the Lipschitz condition while it is simply boundedif m = 0.

Holder spaces, composed of functions satisfying a Holder condition, areespecially relevant for solving partial di!erential equations. In particular, thetopological Holder vector space Ck,m($) consists of all functions f in Ck($),7 Named after the German mathematician Rudolf O.S. Lipschitz (1832-1903).


1.5 Sobolev spaces 51

i.e., having derivatives up to order k, for which the kth partial derivativesare Holder continuous with exponent 0 < m < 1. Furthermore, if the Holdercoe"cient

|f |C0,m(!) = supx,y#!

|f(x)( f(y)|.x( y.m

is finite, then f is said to be uniformly Holder continuous with exponent m in$ and then the Holder coe"cient defines a seminorm, denoted by [·]C0,m(!).Holder spaces C ($) are usually endowed with the norm

.f.Ck,m(!) =$

|"|%k

.D"f.C0(!) +$

|"|=k

[D"f ]C0,m(!) .

where the norm . ·. C0 is classically defined by

.f.C0(!) = supx#!

|f(x)| ,

and the mth-Holder norm is

.f.C0,m(!) = .f.C0(!) + [f ]C0,m(!) .

It can be shown that the Holder space Ck,m($) is a Banach space. It does notmake di!erence whether we use $ or its closure in the previous definitions.

1.5.2 Sobolev spaces of integer order

We remind that a function f # L2($), for an open subset $ " Rn, is identifiedto its associated distribution also denoted by f # D'($) (for the sake ofsimplicity) and the pairing between f and a test function + is written as(f, +) = /f, +0. Likewise, any function f in Ck,m($) can be identified to adistribution.

Notion of weak derivative

The notion of weak derivative allows to generalize the concept of derivative tofunctions that are only integrable, i.e., function s which belong to L1(Omega)space.

Let consider a continuous function f # Ck($), where k is a nonnegativeinteger, % be a multiindex of order |%| = k. Then, if + is a test function,integrating by parts % times leads to

!

!f(x)D"+(x) dx = ((1)|"|

!

!D"f(x)+(x) dx .

We notice that the boundary terms vanishe since the function + is compactlysupported. Furthermore, this identity is stil valid if the function f is onlylocally integrable. But then, the expression D"f has to be considered in thedistributional sense seen before. To this end, we provide the following defini-tion.



Definition 1.5.1 (Weak derivative). Suppose f, g are locally integrablefunctions for some open subset $ and % is a multiindex. The function g issaid to be the %th-weak derivative of f if

!

!f(x)D"g(x) dx = ((1)|"|

!

!g(x)+(x) dx ,

for all test functions + # C!c ($).

If f has a weak derivative, it is written as D"f and its weak derivative is thenuniquely defined up to a zero-measure set. If f is su"ciently smooth to havea continuous partial derivative D"f in the classical sense, then D"f is also apartial derivative of f in the distributional sense. But the converse may notbe true, i.e., D"f may exist in the distributional sense without existing inthe classical sense.

Spaces Hm

The Sobolev space Hm(Rn), for integer m, is defined as the completion (orclosure) of the linear space C!c (Rn) of infinitely di!erentiable functions withcompact support, with respect to the norm

.f.Hm =

(

)$

|"|%m

.D"f(x).2L2(!)

*

+1/2

=

(

)$

|"|%m

!

Rn

|D"f(x)|2*

+1/2

.

We note that the space H0(Rn) coincide with the space L2(Rn).Let $ be a bounded domain in Rn. The space C!($) of infinitely di!er-

entiable functions is not complete with respect to the norm . ·.Hm . Then, thespace Hm($) is defined accordingly as the completion of the space C!($) offunctions that are infinitely di!erentiable in the closure of $ with respect tothe norm

.f.Hm(!) =

(

)$

|"|%m

!

!|D"f(x)|2

*

+1/2

.

The functions in spaces Hm($) have derivatives in the distributional senseof order |%| ' m which belong to Hm*|"|($). It is then worth noticing thatthe space Hm($) consists of functions f # L2($) such that D"f # L2($) forany multiindex % with |%| ' m, i.e.,

Hm($) = {f # L2($) , D"f # L2($) , 0 ' |%| ' m} .

It has the noticeable property of being a Hilbert space8 for the inner productdefined by8 This explains the notation H for such spaces.



(f, g)Hm(!) =$

|"|%m

!

!D"f D"g ,

and the associated norm . · .Hm(!) defined above. The space Hm($) is com-plete since the space L2($) is complete. And for m > s, there is a continuousimbedding Hm($) " Hs($).

For example, we consider the Sobolev space of order 1 on $, denoted byH1($), for the inner product

(f, g)H1(!) =!

!(f(x)g(x) dx + Df(x)Dg(x)) dx ,

and the corresponding norm

.f.H1(!) ="!

!|f(x)|2 dx +

!

!|Df(x)|2 dx

#1/2

.

For m & 1 we define the seminorm

|f |Hm(!) =

(

)$

|"|=m

.D"f.0L2(!)

*

+1/2

,

and we notice that

.f.Hm(!) =2.f.2Hm!1(!) + |f |2Hm(!)

41/2.

The space Hm(Rn) is also interesting as it can be described using Fouriertransform of tempered distributions (cf. Section 1.4.4). A function f # L2(Rn)belongs to Hm(Rn) if and only if (1+|k|2)m/2f(k) # L2(Rn). The norm .f.Hm

is then given by

.f.2Hm = (2")*n

!(1 + |k|2)m/2|f(k)|2 dk .

It is then possible to show that the space S(Rn) is dense in Hm(Rn) and thusthat C!c (Rn) is dense in Hm(Rn), for any integer value m & 0. This justifiesthat Hm(Rn) can be defined as the completion of C!c (Rn) in the norm .·.Hm .

We shall also mention that the spaces Hm provide a convenient way ofattesting the regularity of a function. Here, the regularity (or the degree ofsmoothness) of a function f # Hm correspond to the number of times thefunction f is weakly di!erentiable before its derivatives are no longer in L2.

Sobolev spaces W m,p

For 1 ' p ' ! and m a nonnegative integer, the Sobolev space Wm,p($) iscomposed of all locally integrable real-valued (or complex-valued) functions



f such that for each multiindex |%| ' m, D"f exists in the distributional (orweak) sense in Lp($), i.e.,

Wm,p = {f # Lp($) , D"f # Lp($) , |%| ' m} .

We denote by Wm,p0 ($) the closure of C!c ($) in the space Wm,p($). We have

then the chain of imbeddings

Wm,p0 ($) " Wm,p($) " Lp($) .

We observe that for p = 2, Hm($) = Wm,2($), for every k & 0 and wewill then write classically Hm

0 ($) = Wm,20 ($). Furthermore, we have clearly

W 0,p($) = Lp($).The space W 1,p($) is endowed with the standard norm

.f.W 1,p(!) =2.f.p

Lp(!) + .Df.pLp(!)

41/p

="!

!|f(x)|p + |Df(x)|p dx

#1/p

, if p < !

.f.W 1,"(!) = max(.f.L"(!), .Df.L"(!))

Noticing that Lipschitz continuous functions are di!erentiable a.e., leads toconclude that W 1,!($) = C0,1($).

Likewise, for m > 1, the standard norm on W k,p spaces is defined by

.f.W m,p(!) =

(

)$

|"|%m

.D"f.pLp(!)

*

+1/p

=

(

)$

|"|%m

!

!|D"f |p dx

*

+1/p

.f.W m,"(!) = max|"|%m

.D"f.L"(!) =$

|"%m

ess sup!

|D"f | .

Lemma 1.5.1. Equipped with the standard norms defined above, the spaceWm,p($) is a Banach space. Furthermore, for finite p, Wm,p($) is a separa-ble space. In particular, Wm,2($) is a separable Hilbert space with the innerproduct

(f, g)Hm =$

|"|%m

!

!D"f(x) D"g(x) dx .

The next lemma give some properties of weak derivatives and Wm,p spaces.

Lemma 1.5.2. Consider f, g # Wm,p($) and a multiindex |%| ' m. Then

1. D"f # Wm*|"|,p($);2. (f +µg # Wm,p($) and D"((f +µg) = (D"f +µD"g, for each (, µ # R;3. D&(D"f) = D"+&f , for all multiindices |%| + |*| ' m;4. if - " $ is an open subset, then f # Wm,p(-).



Imbeddings

We have already mentioned that there exist the following imbedding Hm($) "Hs($) for integers m > s. Likewise, Wm,p($) " W s,p($) for m & s and any1 ' p < !. If $ is a bounded set then we know that Lp($) " Lq($) for anyp & q. It follows then that Wm,p($) " W s,q($), whenever p & q and m & s.We have the next results.

Lemma 1.5.3. Suppose $ is a bounded set with a piecewise regular boundary.Then there exist the compact imbeddings

Wm,p($) " Cb($) , for m > n/p

Wm,p($) " Ckb ($) , for m > n/p + k ,

where Cb($) denotes the space of bounded continuous functions on $ withrespect to the supremum norm and Ck

b ($) is the space of functions of Ck($)having bounded derivatives up to order k.

Corollary 1.5.1. There exists a continuous imbedding

Wm,p($) " Lq($) , for m & 0 , m & n

"1p( 1

q

#.

1.5.3 Trace theorems

Next we discuss the notion of trace. Suppose . denotes the piecewise smoothboundary of $. We consider the restriction operator T : C1($) % Lp(. ),Tf = f|! . The problem is that a function f # W 1,p($) is only defined a.e.on $ and since . has n-dimensional Lebesgue measure zero, the restrictionT has no meaning as such. The notion of trace operator has been inventedto overcome this problem. The domain of T is a subset of W 1,p($) and sinceC1($) is dense in W 1,p($), T admits a continuous extension [Ada75].

Lemma 1.5.4 (Trace of W 1,p functions). Suppose $ is bounded and .is C1 and let 1 ' p < !. Then there exists a continuous linear operatorT : W 1,p($) % Lp(. ) such that

1. (Tf)(x) = f(x) for all x # . if f # W 1,p * C0($).2. there exists a constant C > 0 such that

.Tf.Lp(' ) ' C .f.W 1,p(!) ,

for all f # W 1,p($).

The operator T is called the trace operator and Tf is called the trace of fon . .

Corollary 1.5.2. Suppose $ is bounded and . is C1. There exists a contin-uous linear operator T0 : H1($) % L2(. ) such that



1. (T0f)(x) = f(x), for all x # . if f # H1 * C0($).2. there exists a constant C > 0 such that

.T0f.L2(' ) ' C .f.H1(!) , (1.12)

for all f # H1($).

Notice that the trace operator is not defined for functions in L2($).

1.5.4 Fractional order spaces and dual spaces

It is possible to define Sobolev spaces Wm,p($) for arbitrary real number m.In particular, fractional order Sobolev spaces Hm(Rn) can be defined using theFourier transform. Such spaces are of interest when considering for instancethe image of the trace operator defined above, Im(T0). The image is a subsetof L2(. ), dense in L2(. ) that will be denoted H1/2(. ). The space H1/2(. )is a Hilbert space for the norm

.f.H1/2(' ) = infg#H1(!) ,T0g=f

.g.H1(!) .

With this definition, the estimate (1.12) can be improved as follows

.T0f.H1/2(' ) ' .f.H1(!) ,

for all f # H1($). Next, we turn to some particular spaces for which weprovide some properties.

The space H10 ($) can also be defined as follows

H10 ($) = {f # H1($) , T0f = 0} ,

and we observe that H10 (Rn) )= H1(Rn).

Lemma 1.5.5. The space C!c ($) is dense in H10 ($).

By analogy, we consider the subspace of Hm($) defined by

Hm0 ($) = {f # Hm($) , D"f = 0 on .}

for all multiindex |%| < m.An important result, the Poincare-Friedrichs inequality states that the

seminorm

|f |Hm(!) =

(

)!

!

$

|"|=m

|D"f |2 dx

*

+1/2

(1.13)

is a norm on Hm0 ($) equivalent to the norm defined previously

.f.Hm(!) =

(

)!

!

$

|"|%m

|D"f |2 dx

*

+1/2

.

And we have the result



Lemma 1.5.6 (Poincare-Friedrichs inequalities).

1. Suppose $ is bounded in one direction. Then there exists a constant C > 0such that

.f.L2(!) ' C |f |H1(!) , for all f # H10 ($) .

2. If $ is bounded of diameter C (or contained in a cube of side length C),then

|f |Hm(!) ' .f.Hm(!) ' (1 + C)m |f |Hm(!) ,

for all f # Hm0 ($).

We denote by H*1($) the dual space of H10 ($); this space has an impor-

tance in the analysis of partial di!erential equations as will be seen in nextchapter. The definition of H*1($) implies that any function f in H*1($) isa bounded linear functional on H1

0 ($). The space H*1($) is a Hilbert spacefor the dual norm

.f.H!1(!) = supu#H1

0 (!)

"/f, u0

.u.H1(!)

#,

where /·, ·0 denotes the duality pairing between H*1($) and H10 ($).

Since D($) is dense in H10 ($), the functions in H*1($) can be identified

to distributions and this justifies the notation /·, ·0. For this reason, H*1($)is often called a distribution space. For example, given f # L2($) then Df #H*1($)n.

Lemma 1.5.7 (Characterization of H*1). Suppose f # H*1($). The fcan be nonuniquely decomposed as follows

/f, u0 =!

!f0u +

n$

i=1

fi!u

!xidx , with u # H1

0 ($) , (1.14)

where every fi # L2($), for 0 ' i ' n. Furthermore, if f satisfies (1.14) forevery fi # L2($), then we have

.f.H!1(!) = inf

-!

!

n$

i=0

|fi|2 dx

.1/2

,

We will use the notation

f = f0 +n$

i=1

!fi

!xi,

whenever (1.14) holds.Likewise, we denote by H*2($) the dual space of H2

0 ($) and we recallthat

H20 ($) = {f # H2($) , T0f = 0 and T1f = 0} ,



where the trace operator T1 is defined by T1f = Df · n, and n is the outerunit normal to $. We can show that for any f # H2($), T1f is in L2(. ).Since D($) is dense in H2

0 ($), then H*2($) is also a distribution space.We denote by (H1($))' the dual space of H1($). It is a Hilbert space for

the norm.f.(H1(!))$ = sup

u#H1(!),u -=0

f(u).u.H1(!)

.

1.5.5 Generalization of integration by parts formulas

In this paragraph, we remind the classical identities commonly used for writ-ing the weak formulation of partial di!erential equations. From now on, $is supposed to be a bounded domain with Lipschitz continuous boundary de-noted !. . The outer unit normal to $ is defined a.e on !$ and will be denotedby n(x) = (n1, . . . , nn)t(x), i.e. ni is the ith component of n.

Using the density of D($) in H1($), we write the Green formula forfunctions in H1($).

Lemma 1.5.8 (Green’s formula in H1($)). For every f, g # H1($), wehave !

!f

!g

!xidx = (

!

!

!f

!xig dx +

!

(!fgni d/ .

Using the trace operator introduced above, we provide the following result.

Lemma 1.5.9 (Green’s formula in H2($)). For every f # H2($) andg # H1($), the following identity holds

!

!0f g dx = (

!

!Df Dg dx +

!

(!

!f

!ng d/ ,

where 0 denotes the Laplacian operator and !f/!n = Df · n.

And we conclude this section by giving the following result.

Proposition 1.5.1. Suppose f # H1($) such that 0f # L2($). Then, T1fdefines an element of H*1/2($).

1.6 Exercises and Problems


References 59

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60 References

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