an2 lecture notes
DESCRIPTION
Lecture notes for the subject Analysis 2 at the Mathematics department, University of Copenhagen.TRANSCRIPT
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 1 of 85
Notes for Analysis 2 (v 1.0)
These are the personal notes of Anders Munk-Nielsen taken during the lectures in Analysis 2 at the
Department for Mathematical Sciences, University of Copenhagen during the fall of 2010. They are filled with
typos and misunderstandings – ye be warned!
Feel free to email me if you have corrections (I’ll email you the raw docx file then!) or a smart way to get
ripped in 4 days for free.
“Keep it ℝ out there!” – Ghandi
Disclaimer: The notes in this compendium are solely the portrait of the misguided dillusions of yours truly
regarding the subject of mathematical analysis – in particular, the lecturer (Mikael Rørdam), is in no way
responsible for the abundance of errors that will presumably appear here. By reading these notes, you implicitly
accept that said mistakes may defile your own mathematical understanding and that you in that case will
contribute by further spreading the plague so that all students taking the subject will be equally dumb and they
will be forced to lower the required levels for certain grades. Moreover, you accept to be a betteer person and to at
least twice every day will say something nice to someone during your day.
Abstract: The present lecture notes illustrates the imact on a modern individual of being put through an
intense course in abstract mumbo-jumbo. We find that the test subjects were highly susceptible the type of
brainwashing considered in this course. In particular, most subjects were turned into zombies from the ongoing
direct exposition to high levels of mathematical brainwashing.
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 2 of 85
Contents 1 First lecture ............................................................................................................................................... 7
1.1 Theorem 1.5 .......................................................................................................................................... 8
1.2 Something… ......................................................................................................................................... 9
1.2.1 Proof, something about convergent .............................................................................................. 9
1.3 Norms ................................................................................................................................................. 10
1.3.1 Theorem 1.8: Proof that the norm is in fact a norm .................................................................... 10
1.4 Theorem 1.9: Cauchy-Schwartz .......................................................................................................... 11
2 Chapter 1 continued ................................................................................................................................ 12
2.1 Recap .................................................................................................................................................. 12
2.2 Theorem 1.11: Triangular inequality .................................................................................................. 12
2.3 Metric .................................................................................................................................................. 13
2.3.1 Proof of the triangle bandit (3) ................................................................................................... 13
2.4 Theorem 1.13: Parallelogram identity ................................................................................................ 13
2.5 Theorem 1.14: recovering the inner product from the norm ............................................................... 14
3 Chapter 2: Normed spaces ...................................................................................................................... 14
3.1 Example .............................................................................................................................................. 15
3.2 Continuity ........................................................................................................................................... 16
3.2.1 Theorem 2.5: Continuity of addition and scalar multiplication .................................................. 16
3.3 (Linear) subspaces .............................................................................................................................. 17
3.4 Theorem 2.13 ...................................................................................................................................... 17
3.4.1 Examples .................................................................................................................................... 17
4 Lecture 3 ................................................................................................................................................. 18
4.1 Recap .................................................................................................................................................. 18
4.2 Theorem 2.9 ........................................................................................................................................ 19
4.3 Löl ....................................................................................................................................................... 20
4.3.1 Ex 2.11 + thm 2.12 ..................................................................................................................... 20
4.4 Equivalent norms ................................................................................................................................ 21
4.4.1 Equivalent norms ........................................................................................................................ 21
4.5 Theorem 2.13 ...................................................................................................................................... 21
5 Chapter 3: Hilbert and Banach spaces .................................................................................................... 23
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 3 of 85
6 Lecture 4 ................................................................................................................................................. 24
6.1 Let’s roll .............................................................................................................................................. 24
6.1.1 New theorem .............................................................................................................................. 25
6.2 Hilbert spaces ...................................................................................................................................... 26
6.3 The Hilbert space L2 ........................................................................................................................... 26
6.3.1 Reminder: the Riemann integral ................................................................................................. 27
6.3.2 Lebesgue integral........................................................................................................................ 27
6.3.3 Onwards...................................................................................................................................... 29
6.3.4 Summing up ............................................................................................................................... 29
6.4 Fisher’s completeness theorem ........................................................................................................... 29
6.4.1 Equality “almost everywhere” .................................................................................................... 30
7 5th lecture ............................................................................................................................................... 30
7.1 Convexity ............................................................................................................................................ 30
7.2 Theorem; closest point property ......................................................................................................... 31
8 Chapter 4; orthogonal expansions ........................................................................................................... 32
8.1 Definition of orthogonality ................................................................................................................. 32
8.1.1 Examples .................................................................................................................................... 32
8.2 Fourier combo ..................................................................................................................................... 33
8.3 Theorem 4.4; Pythagoras’ theorem ..................................................................................................... 33
8.4 Lemma 4.5 .......................................................................................................................................... 33
8.4.1 Theorem 4.6; a form for the closest point .................................................................................. 34
8.5 Theorem; Bessel’s inequality .............................................................................................................. 34
8.6 Convergence of a series of vectors ..................................................................................................... 35
8.7 Theorem 4.11 ...................................................................................................................................... 35
8.8 Complete, orthonormal sequences. ..................................................................................................... 36
9 Orthonormal sequences ........................................................................................................................... 36
9.1 Theorem 4.4 ........................................................................................................................................ 37
9.2 Theorem 4.15 ...................................................................................................................................... 37
9.3 Hilbert spaces with an orthonormal sequence ..................................................................................... 38
9.3.1 Theorem: isomorphism ............................................................................................................... 38
9.3.2 Theorem 4.19 .............................................................................................................................. 38
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 4 of 85
9.4 Orthogonal complements .................................................................................................................... 39
9.4.1 Theorem 4.22 .............................................................................................................................. 39
9.4.2 Lemma 4.23 ............................................................................................................................... 40
9.5 Theorem 4.24: important theorem. ..................................................................................................... 40
9.5.1 Corollary 4.25 ............................................................................................................................ 41
9.6 Definition 4.26: Direct sum and orthogonal direct sum ...................................................................... 41
10 Convergence in L2 (section 4.2) ............................................................................................................. 42
10.1 Kinds of convergence ..................................................................................................................... 42
10.1.1 Proving uniform => L2 ............................................................................................................... 43
10.1.2 L2 almost implies pointwise ....................................................................................................... 43
10.1.3 Example A .................................................................................................................................. 43
10.1.4 Example B .................................................................................................................................. 44
11 Fourier series ........................................................................................................................................... 44
11.1 Löelenpütz ...................................................................................................................................... 44
11.2 Reminders from AN1 ..................................................................................................................... 45
11.3 Results from AN1 ........................................................................................................................... 46
11.3.1 A remark on where your functions live ...................................................................................... 46
11.4 The new stuff in chapter 5 .............................................................................................................. 46
11.4.1 Idea about how the proof goes .................................................................................................... 47
12 Fourier ..................................................................................................................................................... 48
12.1 Recapping ....................................................................................................................................... 48
12.2 Theorem 5.1 e_n ON basis ............................................................................................................. 48
12.3 Theorem 5.5 (Fejér) ........................................................................................................................ 49
12.3.1 Recalling metic spaces ............................................................................................................... 49
12.4 Proving the griner ........................................................................................................................... 49
12.5 Calculating Fourier coefficients ..................................................................................................... 51
12.6 Proving thm 5.5 .............................................................................................................................. 52
12.6.1 Lemma 0: preeesenting the Fejér Kernel .................................................................................... 52
12.6.2 Lemma 5.2 .................................................................................................................................. 53
12.6.3 Lemma 5.3 .................................................................................................................................. 53
13 Fourier continued .................................................................................................................................... 54
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 5 of 85
13.1 Recapping ....................................................................................................................................... 54
13.1.1 How far did we get in the proof .................................................................................................. 55
13.1.2 LEMMA 5.3 ............................................................................................................................... 56
13.1.3 Theorem 5.5 ................................................................................................................................ 56
13.2 Lemma ............................................................................................................................................ 58
13.3 Thm 5.6 + cor 5.7 ........................................................................................................................... 59
13.4 Theorem 5.8 .................................................................................................................................... 60
13.5 Dual spaces (chapter 6) ................................................................................................................... 60
14 Kap 6 – dual spaces ................................................................................................................................. 61
14.1 Sæt i gnag ....................................................................................................................................... 61
14.2 Theorem 6.3 .................................................................................................................................... 62
14.3 Norm of a bounded linear functional .............................................................................................. 63
14.3.1 Examples .................................................................................................................................... 64
14.4 Combojoe ....................................................................................................................................... 65
14.5 Climax: Thoerem 6.8 (Riesz-Frechét) ............................................................................................ 65
14.5.1 jesus ............................................................................................................................................ 66
15 Ch 7 Operators on Banach and Hilbert spaces ........................................................................................ 67
15.1 Let’s go ........................................................................................................................................... 67
15.2 Thm 7.4 ........................................................................................................................................... 67
15.3 Continuity of linear maps ............................................................................................................... 68
15.4 Various examples ........................................................................................................................... 68
15.5 An operator interpretable as an infinitely dimensional matrix ....................................................... 69
15.6 Example integral operators ............................................................................................................. 70
15.7 Differential operators ...................................................................................................................... 71
15.8 blah ................................................................................................................................................. 72
16 Chapter 7 cont’d ...................................................................................................................................... 73
16.1 Spectrum ......................................................................................................................................... 73
16.1.1 Theorem 7.22 .............................................................................................................................. 73
16.2 Adjoint operator .............................................................................................................................. 74
16.2.1 Theorem (linAlg ......................................................................................................................... 74
16.2.2 Thoerem A** = A ....................................................................................................................... 76
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 6 of 85
16.2.3 Sammensatte operatorer ............................................................................................................. 77
16.3 Hermitian operators ........................................................................................................................ 77
16.3.1 Lemma ........................................................................................................................................ 78
17 On the exam ............................................................................................................................................ 79
18 Overview of the syllabus......................................................................................................................... 79
18.1 Normed spaces ................................................................................................................................ 79
18.2 Inner product spaces ....................................................................................................................... 80
18.2.1 Orthonormal sets ........................................................................................................................ 81
18.2.2 Basis ........................................................................................................................................... 81
18.2.3 Combojuice ................................................................................................................................ 82
18.3 Fourier series .................................................................................................................................. 82
18.4 Linear functional ............................................................................................................................. 83
18.4.1 Dual space .................................................................................................................................. 83
18.5 Operators ........................................................................................................................................ 84
18.5.1 Spectrum ..................................................................................................................................... 84
18.5.2 Adjoint ........................................................................................................................................ 84
18.5.3 Hermitian operators .................................................................................................................... 85
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 7 of 85
1 First lecture
vector space over ℂ
DEFINITION ∷ inner procuct on
- A map; ℂ
- I.e. For all ℂ
- satisfies
- (where means complex conjungation, )
ℂ
NOTE!
ℂ ℝ
Reformulation of
If is a vector space over ℂ with inner product, then is an inner product space.
EXAMPLE ∷ ℂ ℂ
Let the inner product be
Why this definition? → positivity
- See e.g.
(modulus, length of vector)
- and
Scalar?
NOTE! in this setting,
- (since we generally have for ℂ that )
EXAMPLE ∷ infinite dimensional spaces
ℂ
We have
And now we define
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 8 of 85
Let’s look at the axioms that must be met
Actually,
1.1 Theorem 1.5
ℂ
We are assuming that hold.
Then the following statements are true
We start with
use with .
first of all, “⇐” is trivial.
“⇒” assume consider (holds for all, so specially or this)
But we assumed so this is zero.
But then .
EXAMPLE “little ell two”
ℂ
ℂ
(since we think of functions from the natural numbers as series)
NOTE is written “ell”.
DEFINITIONS
ℂ
We now introduce
(another question; ??)
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 9 of 85
1.2 Something…
Suppose , ℂ
DEFINITION
We say, is absolutely convergent if and only if
THEN IT FOLLOWS that is convergent
i.e. ℂ s.t.
“Discount inequalities”
PROOF
first;
second;
∎
1.2.1 Proof, something about convergent
Let’s assume that
(since means that and are square summable, i.e. the infinite sum of squared absolutes is
convergent)
We may now check the axioms…
Now let’s check whether ⇒
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 10 of 85
(fundamental property of taking modules that
1.3 Norms
Let be an inner product space.
Norm, ,
No worry about since the inner product is never negative…
EXAMPLE
Let ℂ ,
Then
EXAMPLE ∷
EXAMPLE ∷ ,
For instance
Since is finite, belongs to , i.e.
⇒
1.3.1 Theorem 1.8: Proof that the norm is in fact a norm
inner pr space,
The first is ok
(ii): by assumption on the inner product…
(iii):
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 11 of 85
- (we take squares so we don’t have to worry about the square root)
FACT ∷
PROOF ∷
now we use the fact that for we have
,
1.4 Theorem 1.9: Cauchy-Schwartz
and
ℂ
PROOF
Assume and not lin.dep.
, show
which is strictly positive for all values of .
We need a trick
- We need to get rid of ℂ
- for … polar decomposition
o think e.g. of , where
Consider now, the special case where has this structure;
ℝ
Note now, that ⇒ since by creation
then
so the equation becomes
This is a quadratic equation in
WE KNOW THAT IT’S POSITIVE! This means that the discriminant must be negative
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 12 of 85
2 Chapter 1 continued
2.1 Recap
vector space over ℂ
inner product on
⇒ for all → associates ℂ
Norm
Defines ∷ ok since
We proved that if
And ℂ ⇒
Cauchy-Schwarz
Also,
Angles between vectors
Then we can define
⇒
so that
NOTE!
- Since we take of , we can’t distinquish between acute and obtuse (spids / stump) angles
- This is because we are taking absolute values
- BUT ∷ otherwise we might risk that was complex and then we wouldn’t know what the angle was
- AHA ∷ this is the price for working with complex numbers.
2.2 Theorem 1.11: Triangular inequality
PROOF
Surprisingly non-trivial to prove.
We will start by squaring
(which we showed last time)
- since and
.
Now use, if ℂ then we can write and then and
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 13 of 85
⇒ this means that which we will use as
where we used Cauchy-Schwarz in the last step,
ERGO ⇒
∎
2.3 Metric
We have to use some axioms for this to be a metric
1. and
2.
3.
2.3.1 Proof of the triangle bandit (3)
now we use a trick,
2.4 Theorem 1.13: Parallelogram identity
Why is this called the parallelogram identity?
(lengths of the arrows = norms of the vectors)
PROOF
(since theorem 1.5 says and )
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 14 of 85
Now we sum them
∎
REMARK ∷ the proof uses the
rewriting which thus requires that be induced by an inner product.
2.5 Theorem 1.14: recovering the inner product from the norm
Or equivalently
so if you now the norm, you can recover the inner product in this way…
PROOF
Let’s go murphys! By expanding, we get;
And
And
where we want to use
And
Now we are ready to sum the equations
3 Chapter 2: Normed spaces
vector spaceover ℂ or ℝ
Let ℝ ℂ denote the field and then let’s look at over the field (Danish: field = “legeme”)
But mostly we’ll be thinking of ℂ.
DEFINITION
A norm on is a function
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 15 of 85
A normed space is a vector space with a norm.
METRIC
The norm gives a metric
We verified earlier that is a metric
PROPERTY ∷ Translation invariance;
PROOF
∎
3.1 Example
Let be a compact metric space, e.g.
Consider continuous functions
ℂ
We could define
NOTE will hold since continuous on the compact space
(⇒ then the supremum theorem states that will have a min/max ≠ ∞)
CLAIM ∷ on
PROOF ∷ of the triangle equality
NOTE ∷
Take
then
Why?
Now we can say
∎
NICE2KNOW ∷ as an exercise, we will show that there is no inner procduct on so that comes from
that norm
(unless consists of only one point)
- IDEA for the proof ∷ doesn’t satisfy the parallelogram identity which any norm induced by an inner
product must (this is used in the proof of the parallelogram identity).
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 16 of 85
3.2 Continuity
RECALL
If we have two metric spaces, and , and we have a function , then we say that
- i.e. for all open subsets of , the “originalmængde” by is an open subset of .
THEOREM (more useful way of thinking about it)
⇒
3.2.1 Theorem 2.5: Continuity of addition and scalar multiplication
normed vector space
i) the addition is continuous
ii) scalar multiplication is continuous
Actually, ii is more difficult than i
PROOF OF ii
Show that then
CLARIFYING
- What does it mean that
- It means that and
- Or that and
Let’s look at the animal and make some tricks
and we’ll prove that this becomes a small number for large
BUT ∷ our assumption is that and
- PROBLEM ∷ we only know that but not that
- SOLUTION ∷
o The map is continuous
o means that ⇒
o ⇒
- THEN
⇒
∎
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 17 of 85
3.3 (Linear) subspaces
Let be a normed vector space over ℂ
ℂ
NOTATION
Let the closure of with respect to metric
DEFINITION ∷ a point is in clos(A) if it is the limit of a sequence entirely in A.
GRINER
In other news
A is closed if in , and , then
Closed linear subspace
DEFINITION , is a closed linear subspace if and is a subspace
3.4 Theorem 2.13
normed vector space. Then all finite dimensional linear subspaces are automatically closed.
3.4.1 Examples
ℝ
what are subspaces?
-
- ℝ
-
, ,
,
-
-
-
o i.e.
- PROPOSITION ∷ is not closed (in fact, , although )
PROOF
- Consider
- CLAIM ∷ (hence not closed)
- We need to look at the norms
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 18 of 85
∎ ( is not closed)
4 Lecture 3
4.1 Recap
A normed space is a vector space over ℂ (or ℝ) with a norm
EX ∷ inner product space then also normed with
- BUT ∷ doesn’t necessarily go the other way, e.g. can’t come from any inner product
- (proof @ doesn’t fulfill parallelogram identity)
EX ∷ Continuous functions
→ gives a metric on .
Then we can define
-
Linear subspace
linear subspace (or just subspace) if
⇒
ℂ ⇒
Closure
→ the closure of
closed subspace if is closed ( ) and is a subspace.
Theorem (an exercise) ∷ All finite dimensional subspaces of a normed space are closed.
- Proof (illustration) ∷ use a basis for and let a subset of this be a basis for , call it
. Let’s consider convergence by , ok by theorem 2.13. , , and
with implies
where for
since and . But then since otherwise we couldn’t have . But
this means that all the coordinates , and this means that can be written as a lincomb
of only . But then . ∎
EXAMPLE ∷ ,
- We saw last time ∷ and
- This shows that is not closed
- CAN BE SHOWN ∷
- In fact ∷
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 19 of 85
EXAMPLE ∷ , with
Here we can define
Now consider
- is a subspace (think about it!)
- Contains 0, sums / scalar prods of functions are also continuous functions (and defined on the
same interval)
- Proposition: is not closed
- Proof @ contradiction
- Consider , which is clearly
- Consider
and note that
- Here, but
- More rigourusly
(where we’ve used;
, for (<∞))
∎ is not closed
EXAMPLE ∷ ,
Here, is closed wrt
- Recall
Note that from the example before,
since the difference evaluated at will always be 1, and this is the largest difference.
CONCLUSION
Different norms will have very different implications for convergence
4.2 Theorem 2.9
subspace, then is (still) a (closed) subspace
PROOF
We must show
- If then
- If ℂ then
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 20 of 85
(it is clear that the closure of the subspace is closed)
FIRST
-
- Then such that and
- Since is a subspace,
- Since “+” is continuous,
- continuity;
- Now, where , so . ∎
4.3 Löl
Let ,
Want to define
- linear subspace generated by (think of )
- closed linear subspace generated by
FACT ∷
i) , family of subspaces of
Then is a subspaces of
ii) family of closed subspaces of
Then is a closed linear subspace of
(fællesmængde af lukkede mængder = lukket)
DEFINE (( THINK if then is the smallest, closed set containing . Put more rigorously))
, family of all subspaces of s.t.
- (define a family which indexes all the sets that contain )
Similaryly, family of all closed subspaces of s.t.
NOW DEFINE
INTUITION
If ⇒ then lin(A) is the smallest linear subspace that conatains
Similarly, if , ⇒ then is the smallest closed subspaces that contains
4.3.1 Ex 2.11 + thm 2.12
i) ℂ
ii)
NOW hear this
Consider
- So has zeros everywhere but 1 on the th coordinate
Now let
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 21 of 85
-
- (not proved, but lecturer’s proposition)
EXERCISE ∷ is a subspace
- is open
- (idea for the proof ∷ if is open
4.4 Equivalent norms
Suppose vector space; and are our two norms (sorry, strange notation ρ)
The norms map and
EXAMPLE ∷ in we had the two norms and
WE SAY
and define the same topology on if they define the same open sets.
Equivalently, and define the same topology on if and only if for all sequences in and all
we have ∷
In conclusion; and define the same topology on iff in ;
EXAMPLE
- Previously, we saw how but
4.4.1 Equivalent norms
and are equivalent if
Note that ⇒ equivalent norms define the same topology (and actually also the converse)
4.5 Theorem 2.13
Any two norms on a finite dimensional vector space are equivalent.
(the example previously didn’t hold because it was infinite dimensional)
PROOF
Choose basis for .
Define “Euclidian norm” on by
ℂ
INDSKUD
- In ℂ we have ℂ
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 22 of 85
-
The proposition is now, any other norm, on is equivalent to
(which means that in respect to convergence, we might always just use instead…
NOTE ∷ some claims in the proof are not actually proven
- e.g. the claim that is a norm on
Let’s now set
Now take
We have to show that
Now we want to use Cauchy-Schwarz
→ it gives us that
⇒ WHICH PROOVES THE FIRST INEQUALITY
We now want show that for some
Define a mapping
ℂ ℝ
NOTICE
- is continuous (proof omitted… to prove it use sequential mappings that are each continuous)
- Given a ⇒
- ⇒ this means that ⇒
NOTICE
- ℂ , such that
ℂ
(basically, is a high-dimensional sphere)
NOTE that is closed and bounded ⇒ is compact
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 23 of 85
Now, put
Now
(since a continuous function achieves max and min on any compact set on which it is defined)
NOW ∷ show for all
a) , ,
⇒
since was the inf over the set.
b) GENERAL CASE
5 Chapter 3: Hilbert and Banach spaces
Inner product spaces and normed spaces are metric spaces.
Workings; ,
or from the inner product
DEFINITION ∷
metric space.
Sequence sequence in .
1. convergent if such that , i.e.
2. is a Cauchy sequence if
⇒
Meaning ∷ the points in the sequence get closer and closer to each other… ⇒ means that “the sequence
really wants to converge” (but doesn’t necessarily)
REMARK
- is convergent ⇒ is Cauchy
DEFINITION
is complete (=fuldstændig) if and only if all Cauchy sequences are convergent.
EXAMPLE
ℝ is complete (with ).
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 24 of 85
Opposed to , which is not complete
- Proof ∷ take a sequence of rational numbers converging to an irrational number
- E.g.
- Here,
- Hence is convergent in ℝ and thus Cauchy. But then it’s Cauchy both in ℝ and .
- BUT since , is only convergent in ℝ.
6 Lecture 4
6.1 Let’s roll
metric space
DEFINITION ∷ in is Cauchy if
⇒
DEFINITION ∷ is complete (fuldstændigt) if all Cauchy sequences are convergent.
EXAMPLE
- ℝ is complete
- is not complete
- IDEA ∷ you have a space, , but it has holes in it , missing values. This is why we invent ℝ
THEOREM (@analysis 1)
- All compact spaces are complete
- (but not opposite, ℝ is not compact!)
THEOREM
complete. .
⇒ implies that any open subsets of the real line are incomplete.
THEOREM
ℂ is complete
PROOF (indication)
- Suppose is Cauchy in ℂ
- Then we can write , ℝ
- Then are Cauchy in ℝ
- since if .
- similarly for
- ℝ is complete
- ⇒ ℝ and
- Now put ℂ
- Then check that and you’re done. ∎
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 25 of 85
6.1.1 New theorem
THEOREM ∷ ℂ and are complete wrt metric coming from norm coming from inner product
ℂ
PROOF for
- Take , a Cauchy sequence in
- so that
, i.e.
- and each ℂ
- Given we can choose so that for all we have
- PROBLEM ∷ find so that
- First, CLAIM;
- For every fixed ;
- The sequence
is Cauchy in ℂ
- Let’s write it out
…
here, each column is Cauchy
- Proof of the claim
take
,
since
and
are Cauchy
∎
- This means, that each column has a limit since ℂ is complete.
- ⇒ ℂ, as
- Then we just define
- BUT we need to check 1) is square summable, 2)
- CLAIM ∷ , i.e. , where
- PROOF ∷ take and look at the sums up to
, since
for
Now make an innocently looking bandit
,
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 26 of 85
Now we have
For
since
IMPORTANT DETOUR ∷ WHY FINITE ?
Q: Why did we only look at a finite ?
A: To be able to interchange and .
TRUE ∷
o (since the sum of two convergent series is convergent to the sum of the
limits)
FALSE ∷
o For one, it’s not sure that the new thing is convergent, and if it is, to the
simple sums.
o Example ∷ , , so
but and so
- CLAIM ∷
- PROOF
- take . ⇒
- Claim ∷ in
- Proof ∷ hence
- ∎
6.2 Hilbert spaces
DEFINITION ∷ A Hilbert space is an inner product space (typically over ℂ) which is complete
- (complete wrt the metric @ the norm @ the inner product)
Example ∷ ℂ , are Hilbert spaces.
Example ∷ , with is not a Hilbert space (i.e. not complete).
- why? because wasn’t closed
- (a subset of a complete space is complete if and only if it is closed)
DEFINITION ∷ A Banach space is a normed space that is complete.
Example ∷ all Hilbert spaces are Banach spaces
- All Hilbert spaces are normed (norm induced from inner prod) and they are complete.
- with is a Banach space.
- (the proof uses that the uniform limit of a continuous function is again continuous)
6.3 The Hilbert space L2
The Hilbert space
EXAMPLE with
- Here, is not complete (problem 3.2)
- SPOILER for the solution;
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 27 of 85
- For each consider;
- for
- is the straight line connecting and in that interval (getting steeper and steeper)
- for .
- Consider for and for
- Problem 3.2 ∷ is Cauchy in but not convergent.
Careful with the second statement.
Each is continuous (albeit pointwise and not differentiable)
is discontinuous and thus can’t be the limit of .
POSSIBLE DEFINITION of
Analogous to ℝ being “ with all the holes”
ℂ
where
and the inner product
DOES IT WORK?
The problem is that
is not always well-defined with the Riemann integral.
→ SOLUTION ∷ Lebesgue proposed a new type of integral.
6.3.1 Reminder: the Riemann integral
The Riemann integral
Integral ∷ area between x-axis and the curve
Idea in Riemann ∷ partition into rectangular aras, so that
where
If is well-behaved (piecewise continuous), then the integral converges.
Example of problematic function, , , , ℝ
The problem ∷ the height, , of the column
6.3.2 Lebesgue integral
Instead of deviding the x-axis, we devide the y-axis
⇒ then the rectangles have fixed height.
- we then just multiply with the length on the x-axis, which is (almost surely, though pathalogicalities)
well-defined
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 28 of 85
where
PROBLEM ∷ do all ℝ have a length?
DEPENDS ∷ on the choice of set theoretical axioms
- Axiom of choice ⇒ possible to create strange subsets that don’t have lengths.
SOLUTION ∷
- Let ℝ be the smallest family of subsets of ℝ satisfying
A ℝ
ℝ ⇒ ℝ ℝ
ℝ ⇒
ℝ
THEOREM ∷ uniqueness of the measure(?)
ℝ
ℝ ⇒
DEFINITION
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 29 of 85
ℂ ℂ ℝ
THEOREM ∷
i) If ℝ is measurable, then
is defined
- (as a Lebesgue integral)
NOTE ∷ If both the Riemann integral and the Lebesgue integral exist, they are equal.
NOTE ∷ if is continuous and is open, then is open and ℝ contains all open subsets of
ℝ by definition.
** IMPORTANT **
- and by this, think e.g. of
ii) If ℂ is measurable and if
then
ℂ is defined (as a
Lebesgue integral)
6.3.3 Onwards
DEFINITION ∷
ℂ
ℂ
REMARK
If are measurable, then is also measurable.
- (recall measurable means that preimages of open sets “are not too bad”)
Then it follows that
(where we’ve used that integrals preserve “order” (inequalities))
These two values are finite if the functions are from , and hence the inner product is defined.
6.3.4 Summing up
We define as the functions that are limits of sequences of functions in by the 2-norm.
6.4 Fisher’s completeness theorem
THEOREM ∷ is complete and hence a Hilbert space.
THEOREM
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 30 of 85
- This means that every element of is a limit of a sequence in
- (which was exactly what we tried to define it as)
- ℂ
-
-
6.4.1 Equality “almost everywhere”
NEW PROBLEM HAS ARISEN
- Consider where for and for
- BUT for , and for
- So and but
NEW NOTION
DEFINITION ∷ almost everywhere
DEFINITION ∷ null set
If ℝ and ( has length 0), then is a null set.
We have
ℝ. is a null set if and only if
⇒
7 5th lecture
Important today ∷ The closest point property
MOTIVATION ∷
- In a Eucledian space, take a closed surface. Then for any given point outside the surface, there is one
unique point on the surface, which is closest to that point.
7.1 Convexity
Let , real or complex vector space.
is convex if and only if for all and , then .
MENTAL PICTURE
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 31 of 85
- the usual line between and on which all elements should remain in (so that e.g. can’t have a
heart’s shape)
7.2 Theorem; closest point property
Let be a non-empty, closed, convex set in a Hilbert space, .
For there is a unique point in , which is closer to than any other point in .
In other words there is a unique s.t. .
FIRST ∷ recall the parallelogram identity
PROOF
Let be a non-empty, closed, convex set in our Hilbert space, .
Let . Then we should find a unique that fulfills the above.
- → since , is finite (since )
For each we may find such that
(just think of as “a small number”, it should work for any ).
CLAIM ∷ is a Cauchy sequence
PROOF
- By the parallelogram law used on we get
Rearranging, we get
since
Using that is convex, , we see that
(with )
So , since was “the smallest possible difference”.
We now get
This proves that is Cauchy, and hence converges to some .
NOW use the property that is closed
⇒ then by definition of closedness.
By definition of , we have that .
Using continuity of , implies that
- (since for all )
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 32 of 85
⇒
Now we prove uniqueness
Suppose ,
By convexity,
, so
.
Applying the parallelogram law to , we get
But by a property of
∎
DISCUSS
Could we exclude some assumptions on ?
- Non-emptyness is needed for to be well-defined.
- Closedness is needed for uniqueness.
- Hilbert space ∷ we used the parallelogram law, which holds for all inner product spaces (and thus for all
Hilbert spaces) but not necessarily for normed spaces (and thus Banach spaces) unless the normed space
is also an inner product space.
It is the closest point property that enables us to work with projections in Hilbert spaces.
8 Chapter 4; orthogonal expansions
8.1 Definition of orthogonality
DEFINITION
Let , inner product space.
If we say that and are orthogonal (written )
A family of non-zero vectors is called an orthogonal system if whenever .
If for all we call it an orthonormal system.
If an orthonormal system can be indexed by , we call it and orthonormal sequence.
- Note that a system inexed by also can be indexed by by “renumbering”.
8.1.1 Examples
In ℂ , the standard basis is an orthonormal system
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 33 of 85
In , is an orthonormal sequence where
Consider (or with an inner product, though )
is an orthonormal sequence where
since
8.2 Fourier combo
For an orthonormal sequence and we call the ’th Fourier coefficient of with respect to
.
The Fourier series of is the formal sum
where we call it “formal” since it doesn’t make sense to add infinitely many vectors.
8.3 Theorem 4.4; Pythagoras’ theorem
If are pairwise orthogonal (i.e. is an orthogonal sequence) in an inner product space, then
IDEA for the proof ∷
- Expand as an inner product ( ) and use linearity.
- Then observe that most terms cancel out.
8.4 Lemma 4.5
Let be an orthonormal system in an inner product space, . Let ℂ and take . Then
where is the ’th Fourier coefficient.
PROOF
By theorem 4.4 (Pythagoras’) we see that
so the rest is just calculations.
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 34 of 85
Let and ’s be fixed and let vary.
Then will cover all of .
Since is fixed, we see that we can only impact on the term .
→ We now deduce from the lemma that the smallest value of occurs when for all .
From this, theorem 4.6 follows
8.4.1 Theorem 4.6; a form for the closest point
Let be an orthonormal system. the closet point of to is .
And the distance, is given as
Corollary
If then showing that , i.e. that is itself the closest point to itself
(of course;)
8.5 Theorem; Bessel’s inequality
If is an orthonormal system in an inner product space, , . Then
(this expression makes sense since all the inner products that we are summing over are just numbers, and we
learned in analysis 1 how to sum infinite series of numbers… first with vectors does it become a problem).
PROOF
For let be the sum
By theorem 4.6,
so
Now let . Then we see that the shit converges.
∎
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 35 of 85
8.6 Convergence of a series of vectors
Let be a normed space and let , (a sequence of vectors).
We say that is convergent with sum , written
if the finite sums converge to (in
norm)
8.7 Theorem 4.11
Let be an orthonormal sequence in a Hilbert space, with ℂ.
Then converges if and only if
PROOF
“⇒”
Suppose is convergent with sum . For we consider
The inner product is continuous, and thus letting we obtain
Bessel’s inequality yields
“⇐”
Suppose
and let .
Pythagoras’ theorem for
(since the tail of an infinite, convergent sum will converge to zero)
Thus is a Cauchy sequence, and since we are in a Hilbert space, it also converges.
∎
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 36 of 85
8.8 Complete, orthonormal sequences.
From Bessel’s inequeality and the theorem we just proved (4.11) the series converges when is
an orthonormal sequence.
Q: But what is the limit? If it is we can say that are the basis vectors and the coordinates.
A: NO! Not in general… need further assumption
→ we make a definition of completeness “so that we really can be sure of this”.
DEFINITION ∷ complete orthonormal sequence.
is complete iff the following holds;
⇒
9 Orthonormal sequences
Given ON (orthonormal) sequence and , we would like
(from last time, makes sense.)
i.e. is convergent in .
EXAMPLE ∷ where the infinite sum doesn’t converge to .
, the standard ON seq, , with 1 on the ’th place.
Let , then the infinite sum with the s as basis doesn’t give the same.
Then is also an ON seq.
For
IN GENERAL
Let us consider
Then
From this we would like to infer that is zero.
DEFINITION
Let ON seq in Hilbert space.
⇒
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 37 of 85
9.1 Theorem 4.4
Let be a complete ON seq. in Hilbert space.
For any we have
and
PROOF
Part 1 already done
Part 2
Use Pythagoras’ theorem (since all vectors in the sum are orthogonal)
Now let and use that is continuous.
∎
9.2 Theorem 4.15
Let be an ON seq in Hilbert space.
TFAE (the following are equivalent)
PROOF
We have already shown (theorems) ⇒ and ⇒
⇒ by contraposition.
Suppose and for all . Then .
But
so is false.
⇒
Suppose and let , . We must prove that .
Let . is a closed subspace of (we can prove it later).
Each and hence (since clin is the smallest linear subspace containing all s and is a
closed linear subspace).
Hence .
But then , so because ⇒ . But this means that (by property of the
inner product). Thus is complete.
∎
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 38 of 85
9.3 Hilbert spaces with an orthonormal sequence
DEFINITION
A Hilbert space is separable if it contains an orthonormal basis inexed by (or finite).
EXAMPLES ∷ ℂ . Actually no others!
DEFINITION
A map between Hilbert spaces is a unitary operator iff it is
- Linear
- Bijective
- Preserves the inner product, i.e. for all .
and are isomorphic iff there is a unitary operator , and we write .
IDEA ∷ the spaces are “almost the same” if we can move from one to another while preserving the structure.
REMARK
That there are no other separable Hilbert spaces than ℂ and simply means that any other will be
isomorphic to those.
9.3.1 Theorem: isomorphism
CLAIM
Let be linear, Hilbert spaces.
Then is unitary iff is surjective and for all .
- This means that bijectivity and preservance of the inner product needs not be checked.
- Surjectivity is checked by finding the vectors that are sent to the zero vector.
PROOF
Polarization identity gives us that if the norm is preserved, so is the inner product.
∎ (wtf?)
9.3.2 Theorem 4.19
Let be a separable Hilbert space. Then is isomorphic to ℂ for some or .
PROOF
When is separable we have a basis that is either finite or infinite
Suppose contains a finite orthonormal basis (ONB), .
For any , is orthogonal to each and hence zero.
Hence, form an algebraic (the usual) basis for .
Hence any can be written with unique constants..
Let ℂ be
ℂ
This is linear and bijective
(think, if you want to hit use , and vectors in have unique representations).
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 39 of 85
ℂ
∎
Suppose contains an ONB . Define by
We have to prove some things
→ first, by Thm. 4.15, , meaning that should be square summable.
→ follows from thm. 4.15.
→ Linearity is obvious.
SURJECTIVITY
- Let be given.
- Consider thm. 4.11: If you have an seq, then converges., i.e. is in .
- And so is surjective.
Thus is unitary, and therefore .
∎
Remarks
Hence, we know all separable Hilbert spaces already – or an isomorphic griner to them.
We will see, that and are isomorphic.
9.4 Orthogonal complements
DEFINE
, is an inner product space.
The orthogonal complement of is
9.4.1 Theorem 4.22
For any set , is a closed (linear) subspace of , inner product space.
PROOF
It is clear that is a subspace.
- If then also since .
It’s also closed.
Let , and assume that . We must prove that .
Let an arbitrary be given, we need to prove that should be zero
since for all .
Thus .
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 40 of 85
This proves closedness.
∎
9.4.2 Lemma 4.23
Let subspace of an inner product space.
Let . Then for alle .
REMARK ∷ this is a characterization of the orthogonal complement.
PROOF
“⇒”
Suppose and . Must show
Then and thus we can use Pythagoras’ theorem
(since )
“⇐”
Suppose for all . Must show that .
Let ⇒ must prove .
For ℂ, (subspace) and so (with “ ”)
Thus, for all ℂ.
Then choose ℂ s.t. and
Let where
Then
Rearrange and divide by , then
where .
This inequality will hold for any .
Hence, we may let and it should still hold.
Hence, the “klemmelemma” gives that for this to be able to hold for all .
Hence, .
9.5 Theorem 4.24: important theorem.
Let be a Hilbert space, closed, non-empty, linear subspace (and thus convex), .
We want to split in two parts – one in and one in .
CLAIM ∷ there are and such that .
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 41 of 85
INTUITION
In ℝ . If - then - .
We can write any as .
PROOF
Inspired by the fantastic “drawing” (intuition above) take to be the closest point of to
(last time we proved that this is possible to choose since is closed, non-empty.)
Then define .
Of course and , but is in ?
For any , so
(because )
Since we see by lemma 4.23 that .
(since defines all vectors in )
∎
9.5.1 Corollary 4.25
Hilbert space, closed linear subspace.
Then .
From the definition, (THINK ABOUT IT)
PROOF
“ ” follows from the definition
“other way”
Let and write where and .
We want to prove that by showing that .
Since ,
⇒
Then which proves that .
∎
REMARK ∷ Important that is closed.
It doesn’t hold in general that
BUT IT ALWAYS HOLDS THAT .
9.6 Definition 4.26: Direct sum and orthogonal direct sum
DEFINITION
Let and be subspaces of a vector space .
Then is the direct sum of and if
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 42 of 85
We write
DEFINITION
If , an inner product space, is the direct sum of and and , i.e. when ,
then we say that is the orthogonal direct sum.
REMARKS
Whenever we have a closed subspace in a Hilbert space, we can split it into a direct sum.
Thus for any closed subspace in a Hilbert space,
cf. one of the theorems we proved earlier.
10 Convergence in L2 (section 4.2)
Convergence in will help us look at Fourier series in chapter 5.
Bounded interval, ,
to this we “knytter”
ℂ
Inner product
We use the 2-norm,
Think of as the completion of the continuous functions.
- In other words, is a dense subset of (wrt. ).
10.1 Kinds of convergence
Let , in
Uniform convergence
uniformly if .
- Where
- Mostly used when are continuous.
- (Definition naturally requires that are bounded. Note that continuous on ⇒ bounded on )
Pointwise convergence
pointwise if , .
Pointwise convergence almost everywhere
pointwise almost everywhere (a.e.) if
is a null-set.
convergence
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 43 of 85
in if
How are they related?
Uniform implies the two others but no other relations hold
- However, ⇒ pointwise almost holds (need a slight modification)
10.1.1 Proving uniform => L2
CLAIM ∷ , THEN
Hence ∷ uniformly ⇒
⇒
PROOF
Now take and you’re done. ∎
10.1.2 L2 almost implies pointwise
THEOREM (MI)
CLAIM ∷ If ,
And in then
IDEA ∷ it doesn’t work for the sequence it self, but the sequence has a subsequence for which it works.
10.1.3 Example A
,
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 44 of 85
pointwise
, so uniformly
, in
10.1.4 Example B
Write
One can show that
Thus,
Fact: so uniformly
Fact: ,
BUT ∷ it is also clear, that if we just choose , then we will have convergence in (towards 0).
11 Fourier series
Recall ∷ series = rækker = uendelige summer.
In analyse 1, we proved pointwise convergence and talked about uniform convergence.
We will now be looking at convergence, which happens for all functions.
11.1 Löelenpütz
Consider ℂ
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 45 of 85
DEFINITION
, let
FACT ∷
PROOF ∷ is continuous and hence measurable. since ,
FACT ∷ is an orthonormal system
PROOF
(find a anti-derivative @ cos/sin)
We have used that
In fact we have,
WE WANT to prove, that is a basis (and that it’s complete)
11.2 Reminders from AN1
Fourier coefficients
ℂ
Bessel’s inequality (AN1 + ch. 4)
(Later, we want to prove that there holds equality)
FACT ∷
QUESTION ∷ Is (anser; yes)
Rephrasing; (since it’s not trivial what the “=” means) to clarify
Put
Q: Does ?
- NOTE ∷ if there is convergence
If yes, what kind of convergence (uni, point, L2)?
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 46 of 85
A: yes, we have convergence
11.3 Results from AN1
THEOREM A (sætning 3.2)
If ℂ is continuous and (i.e. -periodic), then pointwise.
- Moreover, if ℂ is piecewise cts and and , then
i.e. if is discontinuous in , then the Fourier series converges in that point to the average of the limits
from left and right.
THEOREM B
If ℂ is cts and piecewise (e.g. ) and
Then uniformly.
THOEREM C
as in thm. B. Then
(equality in Bessel’s bandit)
11.3.1 A remark on where your functions live
ℂ
ℝ ℂ
CLAIM ∷ and are actually the same.
PROOF ∷ create a bijective mapping between them.
-
- where , so that
∎
(intuition ∷ there is one and only one way to expand a -periodic function to the entire real line)
11.4 The new stuff in chapter 5
THEOREM 5.1
is an orthonormal basis for .
- (we already know that it’s an orthonormal set → the new thing is that it’s complete)
Hence, the following holds
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 47 of 85
(complete = you can’t add another vector to the set so that it’s still an orthonormal set)
( means that is a null-set (e.g. differ only at finitely many points)
11.4.1 Idea about how the proof goes
Theorem 5.5 (Fejér)
If is cts and put
- (the average of the first partial sums)
Then uniformly.
REMARKS
- Previously, we had to assume piecewise
- Thm. 5.5 shows that we can retain uniform convergence if we use the average instead also when
piecewise is not fulfilled.
One can show,
We are going to use thm. 5.5 to prove that 5.1 must follow (that is a basis)
TRICK
We will be importing the following result
THEOREM (MI) ∷ is dense in wrt. .
(hence, such that
- or equivalently, there is a sequence of functions in that converges to .)
COROLLARY is also dense in
- The idea is, we are almost home by using the function, , defined so that for
and . However, instead of just changing the end-point, we change the interval
so that it’s the linear segment that goes to the endpoint (but so that continuity is
maintained). Then the function is still cts. and it converges to our target for .
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 48 of 85
12 Fourier
12.1 Recapping
ℂ
THEOREM ∷ is a Hilbert space.
EASY FACT ∷ is an orthonormal set in
i.e.
.
- TODAY ∷ prove that it is in fact a basis
Fourier coefficients;
- (since )
ℂ are the Fourier coefficients for .
Fourier series for
- Why is the series in ?
- Since Bessel’s inequality gives that
insuring the required by theorem
(something) in the book (which says that converges if .
12.2 Theorem 5.1 e_n ON basis
THEOREM ∷ is an orthonormal basis for .
HneceHence, the following hold (cf. theorem from ch. 4);
E P
- equality in means that they are equal almost everywhere
- (recall that the Fourier series will at the end points be equal to the average of the limits in the end points)
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 49 of 85
(i.e. convergence by two norm)
12.3 Theorem 5.5 (Fejér)
ℝ ℂ is continuous and -periodic.
Now define
- It takes the averages of the fourier coefficients
- By comparison, in you apply weights of , i.e. weight 1 to all coefficients up until and
zero to all larger coefficients.
THEOREM ∷ uniformly as .
RECALL THEOREM (AN1)
- ℝ ℂ coninuous, -periodic and piecewise , then uniformly as .
REMARKS
- NEW ∷ The new theorem doesn’t assume piecewise !
NAJS2KNOW THEOREM (MI)
- is dense in wrt
i.e.
- COOL ∷ think of this as “the definition of ”
- BUT ∷ definition of as set of measurable, finitely square integrable functions makes it a theorem to be
proven.
12.3.1 Recalling metic spaces
Consider metric space, , .
What does it mean ?
⇒ that there exists , , such that .
.
ALSO ∷ is dense (tæt) in iff .
Corrollary
ℂ is also dense in .
12.4 Proving the griner
BOOK ∷ proves is ONB, then says that uniformlly
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 50 of 85
HERE ∷ prove uniformly, then say that (with extras) it follows that is ONB.
CLAIM ∷ COR + THM 5.5 ⇒ THM 5.1
- that is, if uniformly then is an ONB for
PROOF
we show is dense in
⇒ this will imply that by definition of density.
So we have to show that every in can be approximated with by an in our clin.
Enough to show ∷
- , , then s.t. .
Now the following corollary comes in handy
ℂ
HOW?
- For any given we can approximate with a function from
ℂ
We can now apply thm. 5.5;
NOW
- we proved last time that uniform convergence implies convergence in two-norm
⇒
For some , we have
Put
- this is clearly a finite, linear combination of the s
⇒
Moreover,
∎
REMARKS
- VIEW ∷ Fourier as an approximation, e.g. image processing
-
- is approximately remembered by the finite set of coefficients for
large.
- → so instead of sending all pixels in a picture, we could view it as a function, and then transmit it’s
first Fourier coefficients, which might take less resources.
- sequence in ℂ such that
, then
, hence
determines a function.
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 51 of 85
12.5 Calculating Fourier coefficients
EXAMPLE
Hence,
and we see that the Fourier series for is finite and given above.
since
.
Quite easy
EXAMPLE
Fourier coefficients
(use integration by parts)
⇒
Let’s try to apply Parseval’s
And the other side
PARSEVAL now gives
EXAMPLE
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 52 of 85
will hold, but which one?
Take ℝ
(nice function)
Then
L L
Then
L
EXAMPLE
What about the derivative?
⇒
- (since
12.6 Proving thm 5.5
ℝ ℂ is continuous and -periodic.
THEOREM ∷ uniformly as .
PROOF
12.6.1 Lemma 0: preeesenting the Fejér Kernel
Put . Then
where
- (the integral is called a convolution (da: foldning))
- is called the Fejér kernel.
PROOF
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 53 of 85
We start by looking at a part of the sum in
Consider at a point,
gather the bandits
and since is just constant wrt x
Now, we’re ready to take some sums
Then
∎
12.6.2 Lemma 5.2
CLAIM ∷ ℝ , then
PROOF
Tedious calculations.
REMARKS
12.6.3 Lemma 5.3
ℝ
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 54 of 85
REMARKS
- Ad ∷ Note that this means that collapses like a distribution of sorts… For , all the area
under the graph will end up being in the interval no matter ‼
And do note that the total area is constantly
→ so extremely fast for .
PROOF
Clear, since each is -periodic and is made up of these.
Difficult to see that
BUT easy to see that but it isn’t defined as such for
- BUT from the continuity we see that is continuous and then it must also be at
- Here, only contributes when .
∎
13 Fourier continued
13.1 Recapping
Still considering ,
,
,
orthonormal basis
, Fourier coefficients for .
Theorem 5.1 ∷ is an orthonormal basis for
⇒ in particular, this implies
I.e.
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 55 of 85
(If seen in , then we have , since consists of ækvivalensklasser, i.e. = almost everywhere)
or equivalently
or
(since
)
DEFINE
Theorem 5.5 (Fejér)
ℝ ℂ continuous and -periodic. Then
LAST TIME
We proved that Theorem 5.5 + a result from MI ⇒ theorem 5.1
TODAY ∷ we prove theorem 5.5
13.1.1 How far did we get in the proof
LEMMA 0
, ,
Then
where
The proof now reduces to examining the properties of the Fejér Kernel
LEMMA 5.2
ℝ, , then
NOW
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 56 of 85
Now we have two different ways of writing
13.1.2 LEMMA 5.3
ℝ
is easy to see from the sin() expression
is most easy shows from the double-sum expression
has the interpretation that all the area under the graph of , , which is
, will end up being in the interval for any .
PROOF of
Take .
Now make a vurdering
- since for , we have that for
and
∎
13.1.3 Theorem 5.5
, Let
CLAIM ∷
PROOF
NOW ∷ we want to get inside the integral, i.e.
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 57 of 85
CLAIM ∷
PROOF
(using substitution ⇒ and and )
NOW note that
Now use that ℝ ℂ -periodic, then
for any ℝ.
Now we can insert
∎
Thus,
Now, again use -periodicity and integrate over another interval of same length
Now put
ℝ
Since is continuous and is compact, we get that (extreme value theorem?)
uniformly continuous on
- (reminder: Hence ⇒
)
Now we calculate
Hence
⇒
We now want to prove that
.
CLAIM ∷ , ,
First, we use that since
,
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 58 of 85
FIRST ∷ the
NOW ∷ remember how was chosen!
(reminder: Hence ⇒
)
(don’t get confused that now, is also in our interval since
which is equiv to since we consider -periodic functions)
Thus for all we have that .
NOW ∷
Here, uset hat
o Here, perform substitution with , ⇒ . Also ,
⇒
(since we could choose such that
.)
NOW (but it is similar)
This concludes the proof.
∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎∎
13.2 Lemma
Hilber space with orthonormal basis . Let . Then
(which resembles the formula
, just set )
PROOF
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 59 of 85
(since is an ONB)
- We would like to use the linearity of the inner product but that only works for finite sums. But now use
NOW use the continiuityof the inner product (in each variable) (a fact that comes from Cauchy-Schwarz)
∎
13.3 Thm 5.6 + cor 5.7
Suppose or alternatively
, where
Then let or alternatively
,
THEN
AND
ALTERNATIVELY
We have shown that
(where means “isomorphic to”, i.e. “for all practical bandits they’re the same”)
Where
ℂ
WHY?
Since ⇒
and
⇒
AND
the theorem above shows us the connection between the function and the sequence of Fourier coefficients.
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 60 of 85
13.4 Theorem 5.8
Given , then for all and for all
there exists a polynomial , , ℂ,
such that
i.e. all continuous functions can be approximated by polynomials.
PROOF (flavor of it)
Let’s scale so that and and with
(no big deal, just a scaling)
Here, we have that uniformly
THE TASK ∷ approximate with a polynomial!
BEGIN
The definition is
Enough to show that can be approximated by polynomials (the rest is simply linear combinations)
HURRAY! We now that can be approximated by it’s Taylor series!
Here, uniformly on .
∎
13.5 Dual spaces (chapter 6)
Very useful ways of analyzing spaces
Easy to look at for Hilbert spaces
DEFINITION
vector space over ℂ (or ℝ or )
A linear functional on is a linear map ℂ
EXAMPLE
ℂ ℂ
Then consider
ℂ
( ℂ is a Hilber space)
EXAMPLE
ℂ (set of all -touples)
If ℂ, consider ℂ given by
EXAMPLE
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 61 of 85
Hilbert space, , consider ℂ linear functional by
(linearity follows from the fact that the inner product is linear in the first coordinate)
PURPOSE
We want to show that almost any functional is an inner product form.
EXAMPLE (again)
for some
so if
then we must have that
WOW!
14 Kap 6 – dual spaces
Main object of interest in this chapter
THEOREM 6.8 but also 6.3
14.1 Sæt i gnag
DEFINITION ∷ Let vector space over ℂ
A linear functional [lineær functional] is a linear map ℂ.
(a “function” but generalized to come from any vector space )
NOTE ∷ always
EXAMPLE
ℂ , so ℂ
ℂ ℂ
(and is linear)
EXAMPLE
Hilbert space,
ℂ
This is linear since the inner product is linear in the first variable (and is fixed)
EXAMPLE
If we want , then must be
EXAMPLE
,
If then is a linear functional
EXAMPLE
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 62 of 85
is a linear functional.
14.2 Theorem 6.3
POINT ∷ the focus of this course is not just vector spaces but vector spaces with a norm… hence we are interested
in whether linear functional are continuous.
THOEREM 6.3
normed vector space ℂ linear functional
The following are equivalent (the book only has 3, we write 4).
⇒ ℂ
E ⇒ ℂ
where .
REMARKS
- Note, always true that is a linear subspace of by linearity of . The new thing is that this
subspace is also closed.
PROOF
⇒ is Exercise 6.8 (difficult!)
⇒ is trivial (if continuous everywhere, then in particular continuous at )
CLAIM ∷ ⇒
Assume continuous at .
We will use a different definition, namely the - definition of continuoity at .
⇒
True for . Hence ⇒ .
CLAIM ∷ ⇒
(the claim implies that which would finish our proof)
PROOF
Take with . Put .
Then .
We also have
But now we have ⇒ by
∎
∎
CLAIM ∷ ⇒
Assume .
CLAIM
PROOF
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 63 of 85
We know, that if and then (by definition of ).
Now take arbitrary but (if the original claim is already proved since )
Put (ok since )
Then .
Also, .
By since
∎
Show continuous. Note .
by the claim we just proved.
Take a sequence , in by definition.
But since , we now have that which is the def of cont.!
∎
CLAIM ∷
PROOF
Since where is singleton and thus closed.
The original def of continuity was that the preimage of a closed set is also closed.
∎
REMARKS
Definition of preimage
- NOTE ∷ we don’t assume the existence of an inverted function.
NOTE ∷ A linear functional on a normed space is said to continuous or bounded if condition
in theorem 6.3 hold.
- (i.e. bounded = continuous)
14.3 Norm of a bounded linear functional
DEFINITION ∷ If is a bounded linear functional on , then put
THEOREM (=claim)
(where the previous plays the role of in the proof from earlier so we’re just giving it a new name)
DEFINITION normed vector space.
VERY important thing!
Here, we will only be interested in the dual space of a Hilbert space.
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 64 of 85
THOEREM
is a Banach space
( )
ℂ ⇒
the book also proves that the norm on this space is in fact a norm.
14.3.1 Examples
Hilber space,
ℂ
(where the inequality is by Cauchy-Schwarz, my homies)
Hence is bounded (continuous)
CLAIM ∷
PROOF
We already have
Assume
Put so that .
Then since is a particular vector of length 1 and is sup over such vectors
∎
EXAMPLE
, ,
Is bounded in
1. ?
2. (
Let’s look
1. ,
This shows .
It is also true that
(consider . , ).
IMPORTANT COMMENT
- He just said that is a Banach space but is not!
- Recall the difference from measurability and spaces where (perhaps) it is the converse (??!?!?!?!)
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 65 of 85
14.4 Combojoe
FACT ∷ Hilbert space, closed subspace, , then
RECALL ∷
Why the fact?
, hence ⇒
Another way, , then for and .
Pick , then .
- Otherwise and then .
14.5 Climax: Thoerem 6.8 (Riesz-Frechét)
Let be a Hilbert space. The following holds
Let be a bounded linear functional on
where
This means that every bounded linear functional on are of the inner product form
⇒
and
Hence, the map , satisfies
meaning that we can think of it as “ ”
- but unfortunately is only conjugated linear and not linear.
REMARKS
We’ve already proved .
PROOF
is already proved
CLAIM holds
PROOF
First we need theorem 1.5(iv): ⇒
, so that
(easy proof, follows from 1.5(iv))
∎
CLAIM holds
PROOF
Take bounded linear functional
CASE 1
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 66 of 85
, now take and we’re done
CASE 2
. Then is a closed linear subspace of .
Since , . Then so that where .
o (note
, .
Put
.
Then
Now we have found a vector of length 1 in the orthogonal complement to .
Now take , and note ℂ is just a number.
consider
Hence, .
This means that .
Hence
And deviding yields
∎
REMARK
Lecturer “THIS IS THE MOST FUNDAMENTAL PROOF OF THINKING” (w00t??)
14.5.1 jesus
EXAMPLE
,
(not a Hilbert space)
ℂ
Easy to see linear functional on .
Bounded?
- Well, we can see that for some
- → if we just set .
Huzzah, this instantly gives us that
And we now that which helps
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 67 of 85
15 Ch 7 Operators on Banach and Hilbert spaces
Two main reasons for studying Hilbert spaces; Fourier and Operators.
Operators on Hilbert spaces; motivated as a correct mathematical formulation of Quantum Mechanics
15.1 Let’s go
vector spaces over ℂ.
A mapping, is linear if
ℂ
NOTE ∷ we often write , i.e. we almost view it as a product of and .
EXAMPLE
ℂ , ℂ ℂ is linear if and only if for some matrix
Thus,
15.2 Thm 7.4
Suppose and are normed spaces and is linear. The following are equivalent;
REMARK
- The new thing in this course is that we work with metrics often (typically derived from the norm)
DEFINITION
is linear, then is bounded if
(we call the operator norm of .)
REMARK
- It looks like a theorem from last time (which also included that the kernel of something was closed)
i.e. is not necessarily closed.
PROOF
Exactly analogous to that from last time.
∎
FURTHER
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 68 of 85
If is bounded, then
GENERAL REMARK on notation
and … it all really depends on the input for the norm.
The subscript merely emphasizes the obvious, that the norm must correspond to it’s input.
15.3 Continuity of linear maps
Are all ℂ ℂ continuous?
→ YES! Because ℂ is finitely dimensional
- (can be proven to hold for any finitely dimensional vector space)
Onwards
Given ℂ ℂ , matrix. What is ?
- No simple formula
- One (stupid) estimate: (since is finite)
15.4 Various examples
EXAMPLE (i)
ℂ ℂ
Let
Here,
CLAIM ∷
PROOF
- First,
-
. Hence .
- Now,
- Tak ℂ arbitrary, ℂ
-
- Hence,
This gives . ∎
EXAMPLE (ii), the Fibonacci map
ℂ ℂ
By (the stupid estimate),
CLAIM ∷
the golden value
REMARK
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 69 of 85
- Eigenvalues of ;
,
.
EXTRA REMARK
There are eigenvectors, ℂ, such that and .
AND such that is an orthonormal basis for ℂ .
- The reason is that is symmetric (or rather, something-something complex symmetric, but that is
just like being symmetric when all elements are real)
PROOF (incomplete… similar to the previous)
⇒
take ℂ. Since is a basis, we can write any such vector as
THE RESULT ∷ the maximum eigenvalue must be the norm of .
EXAMPLE (iii)
ℂ ℂ
gives that
Eigenvalues of are (note that is upper triangular)
All eigenvectors are and .
NOTE ∷ then the eigenvectors do not represent an orthonormal basis and hence is not equal to the
largest eigenvalue.
15.5 An operator interpretable as an infinitely dimensional matrix
,
, linear.
means that , hence, ℂ such that is measurable and
.
Now we want to define . We want it to satisfy
-
- ℂ
- (pointwise multiplication, )
CLAIM ∷ linear, bounded and .
PROOF
LINEAR
One must show .
Both sides are functions, so we evaluate at arbitrary , .
Clearly measurable (?!)
But finite integral?
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 70 of 85
Hence, .
Hence .
NOTE
linear and st then
BECAUSE .
EXAMPLE
, . Note that .
Hence,
Show .
i.e. find , , .
→ this turns out to be impossible!
But a little less might do… if we can only find , and
→ since then we would just take the and get what we want (sup = 2 ≥ 2).
Take
→ one can now check that
.
Now see that
But this proves so that
15.6 Example integral operators
Recall the Fejér kernel,
PURPOSE NOW
- Input = , output .
- This is actually a linear map.
More generally,
Consider the two intervals and in ℝ
Consider the function
ℂ
And
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 71 of 85
And the function at a given value, is
- Fejér kernel case is an example of this approach:
- Here
-
- Then
Returning to the general case, we want to show that such a “kernel operator” is always linear and bounded
CLAIM
If then . Moreover, linear, bounded and
PROOF
skipped here… the book has all the details…
∎
15.7 Differential operators
EXAMPLE
Define the operator ℝ ℝ (the book write )
We want to write .
However, this is not generally possible since ℝ contains many functions which are not differentiable.
- and even if it is, it’s not sure that ℝ .
INSTEAD, let domain of ℝ ℝ .
- NOTE ℝ (actually dense)
Clearly, is linear ( )
BUT, is unbounded; .
CLAIM is unbounded; .
PROOF (outline)
Note,
(since we get an “ ” down when we differentiate)
⇒
∎
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 72 of 85
15.8 blah
EXAMPLE
Again, we must restrict us to the domain of ,
Then , is a dense subspace.
Consider
We can easily see
Hence,
Aha, so for all , is an eigenvalue for with eigenvector .
Moreover, we see that the set of eigenvectors, , is an orthonormal basis for .
NOW
Why does this show that is unbounded?
It now follows that
∎
Finally,
⇒
⇒
Wow, have we now defined the derivateive of any function?
→ well not exactly, it only works when
But this allows us to redefine the set .
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 73 of 85
16 Chapter 7 cont’d
16.1 Spectrum
Banach space (or Hilbert space)
ℂ
ℂ ℂ ⇒ A
ℂ ℂ
(where .)
EXAMPLE
In physics, the spectrum of a operator is “the set of numerical observations for that operator”.
16.1.1 Theorem 7.22
Banach space,
⇒
equivalently, (Kugle = Ball)
Ingredients for the proof
⇒
⇒
We showed these last time.
PROOF
Consider is closed (omvendte af åben)
Now define
ℂ
is a continuous mapping.
Now we can define the spectrum of
ℂ ℂ
∎
PROOF
Take ℂ such that . Enough to show that This implies that .
(showing that the negation of RHS implies the negation of LHS)
Here, note that
By this show that is invertible. Hence it is also when multiplied by .
⇒
⇒
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 74 of 85
EXAMPLE
16.2 Adjoint operator
More generally
16.2.1 Theorem (linAlg
ℂ ℂ linear,
ℂ
Hermitian (self-adjoint) case, .
THEOREM
Hilbert spaces,
Then
SPECIAL CASE
, then it is formulated as .
MOREOVER
LEMMA (not in book)
Let . Then
PROOF of lemma
Let ,
put and note
(note that so )
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 75 of 85
Hence,
Since was defined to be the sup of such ’s when was also allowed to move freely.
∎
PROOF OF THE THEOREM existence
(use the fact that ⇒ .)
∎
PROOF OF THE THEOREM of the existence of the adjoint operator
Take and define
ℂ
is linear since is linear and is linear in the first variable.
CLAIM ∷ is bounded
PROOF
Hence,
(sicne )
qed
By Riesz-Frechet such that
Put
Then
Hence
So we must show that is linear and that is bounded (and then we have proved existence)
(SO FAR ∷ we have proved that a mapping exists… we need to prove that it’s linear and bounded)
CLAIM is linear
PROOF
Take , ℂ (we still have )
Look at the following
We now want to conclude that this implies that
This is true because (chapter 1): ⇒
qed
CLAIM is bounded
PROOF
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 76 of 85
(now use , and )
qed
∎
Example
CLAIM ∷
PROOF
. Show (which implies )
From which we see that .
∎
ANOTHER EXAMPLE
CALAIM
PROOF
Similar to above, write out each side, then they are both equal to
and hence is an adjoint operator and by the theorem from before, it is the unique.
16.2.2 Thoerem A** = A
THEOREM
REMARKS
Note that . .
PROOF
Take and and look at and use that is the adjoint.
Hence, for all , so by thm. from chapter 1
⇒
we have that
⇒
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 77 of 85
∎
16.2.3 Sammensatte operatorer
sets,
Then
(note that and not )
THEOREM ∷
PROOF
Take .
Now use the defining equation for the adjoint operator of
Hence, is the adjoint operator of .
By the theorem from earlier, it is the only adjoint operator, i.e.
∎
THEOREM
, ℂ, then
(i.e. the adjoint-operator is conjungated linear)
16.3 Hermitian operators
Definition
Hilbert space, .
Then is Hermetian (da: hermitesk) or self-adjoint (da: selv-adjungeret) if
REMARKS
In many senses, in a complex world, being hermitian means that you are real (have imaginary part zero)
Also, you can make spectral theory from it (analogous to diagonalizing)
EXAMPLE
ℂ . Which of the following are Hermetian?
- is
- is not
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 78 of 85
- is not
- is.
EXAMPLE
We showed that
But now we can state the following
ℝ
i.e. that being hermetian is equal to being real.
INSHOT (indskud)
Take ℂ, then , ℝ.
EXAMPLE
, put
,
CLAIM and are hermetian and .
PROOF
Easy to see that .
Take . Now use that is conjugated linear
∎
REMARK
The conclusion is that if you have an operator, , that is not hermetian, then you can create a hermetian
operator from it by using this recipe.
16.3.1 Lemma
LEMMA , complex Hilbert space, then
⇒
NOTE! This does not hold for a real Hilbert space (åmgwtf?)
THEOREM ∷ , then
ℝ
THEOREM ∷ , hermetian, then
ℝ
EXAMPLE
, multiplication operator
The thing is
ℝ
We can almost see this fact because
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 79 of 85
so for some . (proves the theorem for multiplication operators)
INDICATION OF WHY THIS HERE HOLDS
PROOF OF THEOREM ℝ
“⇒”
Assume .
Now since ⇒ ℝ, this means that
ℝ
“⇐”
, , . Show that if ℝ.
We assume that ℝ and want to show that .
This is so because for ,
Now by the first part of the proof, ℝ and ℝ.
But we now that ℝ by the assumption and hence the imaginary part of this must be zero.
By lemma, this means that and hence . Since , .
∎
17 On the exam
If you write in hand, he recommends using a kuglepen end not a blyant.
We are allowed to use facts stated in the book even if there’s no proof for them (if they’re in a bisætning)
We are allowed to use problems that were proved during exercises.
- However, in the true/false questions, less argument is required – sometimes even just stating “false; we
proved this in an exercise”.
18 Overview of the syllabus
18.1 Normed spaces
vector space over ℂ with
⇒ then we get a metric, .
In this sense, gets a topology (open/closed sets, continuity)
EXAMPLE ∷ subspaces
- We have subspaces, but in particular we can talk about open and closed subspaces
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 80 of 85
- Not all subspaces are closed,
- e.g. is not closed (i.e. taken in is not equal to )
EXERCISE
- Normed space, finite dimensional subspace, then is closed.
COMPLETE SPACE ∷ complete if all Cauchy sequences are convergent.
BANACH SPACE ∷ is a Banach space if normed space and complete.
18.2 Inner product spaces
INNER PRODUCT SPACES ∷ vector space over ℂ with inner product ℂ.
Linear in frist variable, conjugate linear in second variable.
NOTE ∷ inner product norm metric
HILBERT SPACE ∷ Inner product space that is complete.
CAUCHY SCHWARZ
- Also, equality holds if and only if for a ℂ.
EXAMPLES of inner product spaces
- ℂ , , ,
- In ℂ ∷
- Hilbert space
- In ∷ with
- (and this will converge since ⇒
-
- Hilbert space
- In ∷
- with norm
- Not Hilbert space
I.e. there are Cauchy sequences in that aren’t convergent.
- ℂ
- Contains but also piecewise functions and even more exotic functions.
- Intuitive understanding ∷ is the completion of with respect to .
i.e. “take all Cauchy sequences in and if their limit is not in , include it in
as well”.
Sort of like ℝ is the completion of .
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 81 of 85
I.e. all functions in is the limit of a sequence of functions in
- Hilbert space
18.2.1 Orthonormal sets
Relevant in Hilbert spaces.
ORTHONORMAL SET ∷
- Let be a Hilbert space. Let , .
- Then is an orthonormal set if
. I.e. and .
FINITE CASE ∷
INFINITE CASE ∷ or
- (instead of using an arbitrary index set, , we will be using the natural numbers)
EXAMPLE
- ℂ , here the orthonormal basis is
- Since we may write ℂ as .
- Orthonormal system (even ON basis)
- ,
- ,
- ∷
- is an orthonormal set (even ON basis)
18.2.2 Basis
COMPLETE ∷ An orthonormal set is complete if
- i.e. is the maximal wrt being an ON set
ORTHONORMAL BASIS ∷ is an orthonormal basis for if it is an ON set and is complete
THEOREM
Let be a Hilbert space, and let be an orthonormal set. Then the following are equivalent
ON
P
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 82 of 85
- By we mean that
18.2.3 Combojuice
inner product space (⇒ )
PARALLELOGRAM IDENTITY
THEOREM ∷ Closest point property (proved using parallelogram identity)
Hilbert space, closed and conved. Then
- We say that “ is the closest point in to ”
- Only works in Hilbert spaces (not Banach) since the parallelogram identity is used.
THEOREM 4.6
Hilbert space, orthonormal set in .
Put (which is also the closed linear span(!))
Let . Then the closest point, , in to is given by
ORTHOGONAL COMPLIMENT ∷ Let closed subspace. Then define
⇒
THEOREM
- It works for all closed subsets, in particular subspaces
- ⇒ hence it can be used “somehow” in relation to the closest point theorem…
(didn’t quite hear that)
18.3 Fourier series
THEOREM 5.1 ∷ is an orthonormal basis for
- (with ∷
COR ∷ ,
.
- NOTE! There may be confusion!
- Book:
- Lectures:
COR ∷ , then
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 83 of 85
Put , then we are saying that
or
We also say
PRACTICAL REMARK
- Finding coefficients can be tedious since the inner product involves an integral
- But for some nicer functions, it will be doable
THEOREM ∷ Fejér
- , (or ℝ ℂ continuous and -periodic)
- Then for
we have
18.4 Linear functional
normed vector space.
LINEAR FUNCTIONAL on ∷ ℂ.
Let ℂ linear functional. Then we may define
OPERATOR NORM
NOTE ∷
BOUNDED OPERATOR ∷ is bounded if
THEOREM
- If ℂ linear functional, then the following are equivalent
18.4.1 Dual space
DUAL SPACE ∷ the dual space to is
is a Banach space
- This holds even if itself is not complete
EXAMPLE
- Hilbert space, . Then we can define
ℂ
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 84 of 85
- From Cauchy-Schwarz (more or less) we get
- Since C-S gives and compare it with
THEOREM (Riesz-Frechét)
- Hilbert space, bounded linear functional ℂ,
- Then
- Or more precisely
- (and )
- Also, ⇒
18.5 Operators
normed spaces
If linear, we can (in analogy to the functional norm) define the operator norm
BOUNDED ∷ is bounded
THEOREM ∷ bounded continuous
Notation ∷
Relation:
Sammensatte operatorer
- then we can associate
- We get that
INVERTIBILITY
- invertible if and
- Where ∷
18.5.1 Spectrum
then we define
ℂ ℂ
Properties of the spectrum
-
- closed
- ⇒
- Generalizes the notion of eigenvalues from linear algebra
18.5.2 Adjoint
Hilbert spaces,
⇒
Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010
Page 85 of 85
- We used Riesz-Frechét to prove this
PROPERTIES
-
-
-
-
18.5.3 Hermitian operators
Hilbert space,
ℝ ℝ
Consider the Hilbert space and
then we associate an operator by