An2 Lecture Notes

Notes for Analysis 2 (v 1.0) Anders Munk-Nielsen October 2010

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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.


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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


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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


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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


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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


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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


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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


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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; ??)


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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 ⇒


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(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):


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- (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


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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


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⇒ 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 )


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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


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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).


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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

⇒

∎


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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


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∎ ( 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 ∷


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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


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(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


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-

- (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


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-

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


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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 ).


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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. ∎


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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

,


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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;


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- 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


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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


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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


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- 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


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- 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 )


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⇒

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


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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.


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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.

∎


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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.

∎


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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.

⇒


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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.

∎


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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).


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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 .


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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 .


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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


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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


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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

,


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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


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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)?


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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


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(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 .


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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)


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(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


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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.


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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


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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


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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


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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.


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(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


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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.


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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

,


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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


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(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.


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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


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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


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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


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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.


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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 (??!?!?!?!)


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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


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, 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


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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


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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


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- 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?


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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


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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)

⇒

∎


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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 .


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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 .

⇒

⇒


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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 )


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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


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(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

⇒


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∎

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


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- 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


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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


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- 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 .


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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


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- 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


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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


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- 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,

⇒


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- 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

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