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Lecture notes for the subject Analysis 2 at the Mathematics department, University of Copenhagen.
Notes for Analysis 2 (v 1.0)
Anders MunkNielsen
October 2010
Notes for Analysis 2 (v 1.0)These are the personal notes of Anders MunkNielsen 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 (Ill email you the raw docx file then!) or a smart way to get ripped in 4 days for free. Keep it 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 Rrdam), 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 mumbojumbo. 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.
Page 1 of 85
Notes for Analysis 2 (v 1.0)
Anders MunkNielsen
October 2010
Contents1 1.1 1.2 First lecture ............................................................................................................................................... 7 Theorem 1.5 .......................................................................................................................................... 8 Something ......................................................................................................................................... 9 Proof, something about convergent .............................................................................................. 9
1.2.1 1.3
Norms ................................................................................................................................................. 10 Theorem 1.8: Proof that the norm is in fact a norm.................................................................... 10
1.3.1 1.4 2 2.1 2.2 2.3
Theorem 1.9: CauchySchwartz.......................................................................................................... 11 Chapter 1 continued ................................................................................................................................ 12 Recap .................................................................................................................................................. 12 Theorem 1.11: Triangular inequality .................................................................................................. 12 Metric .................................................................................................................................................. 13 Proof of the triangle bandit (3) ................................................................................................... 13
2.3.1 2.4 2.5 3 3.1 3.2
Theorem 1.13: Parallelogram identity ................................................................................................ 13 Theorem 1.14: recovering the inner product from the norm ............................................................... 14 Chapter 2: Normed spaces ...................................................................................................................... 14 Example .............................................................................................................................................. 15 Continuity ........................................................................................................................................... 16 Theorem 2.5: Continuity of addition and scalar multiplication .................................................. 16
3.2.1 3.3 3.4
(Linear) subspaces .............................................................................................................................. 17 Theorem 2.13 ...................................................................................................................................... 17 Examples .................................................................................................................................... 17
3.4.1 4 4.1 4.2 4.3
Lecture 3 ................................................................................................................................................. 18 Recap .................................................................................................................................................. 18 Theorem 2.9 ........................................................................................................................................ 19 Ll ....................................................................................................................................................... 20 Ex 2.11 + thm 2.12 ..................................................................................................................... 20
4.3.1 4.4
Equivalent norms ................................................................................................................................ 21 Equivalent norms ........................................................................................................................ 21
4.4.1 4.5 5
Theorem 2.13 ...................................................................................................................................... 21 Chapter 3: Hilbert and Banach spaces .................................................................................................... 23 Page 2 of 85
Notes for Analysis 2 (v 1.0) 6 6.1
Anders MunkNielsen
October 2010
Lecture 4 ................................................................................................................................................. 24 Lets roll.............................................................................................................................................. 24 New theorem .............................................................................................................................. 25
6.1.1 6.2 6.3
Hilbert spaces...................................................................................................................................... 26 The Hilbert space L2........................................................................................................................... 26 Reminder: the Riemann integral ................................................................................................. 27 Lebesgue integral........................................................................................................................ 27 Onwards...................................................................................................................................... 29 Summing up ............................................................................................................................... 29
6.3.1 6.3.2 6.3.3 6.3.4 6.4
Fishers completeness theorem ........................................................................................................... 29 Equality almost everywhere .................................................................................................... 30
6.4.1 7 7.1 7.2 8 8.1
5th lecture ............................................................................................................................................... 30 Convexity............................................................................................................................................ 30 Theorem; closest point property ......................................................................................................... 31 Chapter 4; orthogonal expansions ........................................................................................................... 32 Definition of orthogonality ................................................................................................................. 32 Examples .................................................................................................................................... 32
8.1.1 8.2 8.3 8.4
Fourier combo ..................................................................................................................................... 33 Theorem 4.4; Pythagoras theorem ..................................................................................................... 33 Lemma 4.5 .......................................................................................................................................... 33 Theorem 4.6; a form for the closest point .................................................................................. 34
8.4.1 8.5 8.6 8.7 8.8 9 9.1 9.2 9.3
Theorem; Bessels inequality.............................................................................................................. 34 Convergence of a series of vectors ..................................................................................................... 35 Theorem 4.11 ...................................................................................................................................... 35 Complete, orthonormal sequences. ..................................................................................................... 36 Orthonormal sequences ........................................................................................................................... 36 Theorem 4.4 ........................................................................................................................................ 37 Theorem 4.15 ...................................................................................................................................... 37 Hilbert spaces with an orthonormal sequence..................................................................................... 38 Theorem: isomorphism ............................................................................................................... 38 Theorem 4.19.............................................................................................................................. 38 Page 3 of 85
9.3.1 9.3.2
Notes for Analysis 2 (v 1.0) 9.4
Anders MunkNielsen
October 2010
Orthogonal complements .................................................................................................................... 39 Theorem 4.22.............................................................................................................................. 39 Lemma 4.23 ............................................................................................................................... 40
9.4.1 9.4.2 9.5
Theorem 4.24: important theorem. ..................................................................................................... 40 Corollary 4.25 ............................................................................................................................ 41
9.5.1 9.6 10
Definition 4.26: Direct sum and orthogonal direct sum...................................................................... 41 Convergence in L2 (section 4.2) ............................................................................................................. 42 Kinds of convergence ..................................................................................................................... 42 Proving uniform => L2............................................................................................................... 43 L2 almost implies pointwise ....................................................................................................... 43 Example A .................................................................................................................................. 43 Example B .................................................................................................................................. 44
10.1 10.1.1 10.1.2 10.1.3 10.1.4 11
Fourier series........................................................................................................................................... 44 Lelenptz ...................................................................................................................................... 44 Reminders from AN1 ..................................................................................................................... 45 Results from AN1 ........................................................................................................................... 46 A remark on where your functions live ...................................................................................... 46 The new stuff in chapter 5 .............................................................................................................. 46 Idea about how the proof goes.................................................................................................... 47
11.1 11.2 11.3 11.3.1 11.4 11.4.1 12
Fourier ..................................................................................................................................................... 48 Recapping ....................................................................................................................................... 48 Theorem 5.1 e_n ON basis ............................................................................................................. 48 Theorem 5.5 (Fejr) ........................................................................................................................ 49 Recalling metic spaces ............................................................................................................... 49 Proving the griner ........................................................................................................................... 49 Calculating Fourier coefficients ..................................................................................................... 51 Proving thm 5.5 .............................................................................................................................. 52 Lemma 0: preeesenting the Fejr Kernel .................................................................................... 52 Lemma 5.2 .................................................................................................................................. 53 Lemma 5.3 .................................................................................................................................. 53
12.1 12.2 12.3 12.3.1 12.4 12.5 12.6 12.6.1 12.6.2 12.6.3 13
Fourier continued .................................................................................................................................... 54 Page 4 of 85
Notes for Analysis 2 (v 1.0) 13.1 13.1.1 13.1.2 13.1.3 13.2 13.3 13.4 13.5 14
Anders MunkNielsen
October 2010
Recapping ....................................................................................................................................... 54 How far did we get in the proof.................................................................................................. 55 LEMMA 5.3 ............................................................................................................................... 56 Theorem 5.5................................................................................................................................ 56 Lemma ............................................................................................................................................ 58 Thm 5.6 + cor 5.7 ........................................................................................................................... 59 Theorem 5.8 .................................................................................................................................... 60 Dual spaces (chapter 6)................................................................................................................... 60
Kap 6 dual spaces................................................................................................................................. 61 St i gnag ....................................................................................................................................... 61 Theorem 6.3 .................................................................................................................................... 62 Norm of a bounded linear functional .............................................................................................. 63 Examples .................................................................................................................................... 64 Combojoe ....................................................................................................................................... 65 Climax: Thoerem 6.8 (RieszFrecht) ............................................................................................ 65 jesus ............................................................................................................................................ 66
14.1 14.2 14.3 14.3.1 14.4 14.5 14.5.1 15
Ch 7 Operators on Banach and Hilbert spaces ........................................................................................ 67 Lets go ........................................................................................................................................... 67 Thm 7.4........................................................................................................................................... 67 Continuity of linear maps ............................................................................................................... 68 Various examples ........................................................................................................................... 68 An operator interpretable as an infinitely dimensional matrix ....................................................... 69 Example integral operators ............................................................................................................. 70 Differential operators ...................................................................................................................... 71 blah ................................................................................................................................................. 72
15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 16
Chapter 7 contd...................................................................................................................................... 73 Spectrum ......................................................................................................................................... 73 Theorem 7.22.............................................................................................................................. 73 Adjoint operator.............................................................................................................................. 74 Theorem (linAlg ......................................................................................................................... 74 Thoerem A** = A....................................................................................................................... 76 Page 5 of 85
16.1 16.1.1 16.2 16.2.1 16.2.2
Notes for Analysis 2 (v 1.0) 16.2.3 16.3 16.3.1 17 18
Anders MunkNielsen
October 2010
Sammensatte operatorer ............................................................................................................. 77 Hermitian operators ........................................................................................................................ 77 Lemma ........................................................................................................................................ 78
On the exam ............................................................................................................................................ 79 Overview of the syllabus......................................................................................................................... 79 Normed spaces................................................................................................................................ 79 Inner product spaces ....................................................................................................................... 80 Orthonormal sets ........................................................................................................................ 81 Basis ........................................................................................................................................... 81 Combojuice ................................................................................................................................ 82 Fourier series .................................................................................................................................. 82 Linear functional............................................................................................................................. 83 Dual space .................................................................................................................................. 83 Operators ........................................................................................................................................ 84 Spectrum..................................................................................................................................... 84 Adjoint ........................................................................................................................................ 84 Hermitian operators .................................................................................................................... 85
18.1 18.2 18.2.1 18.2.2 18.2.3 18.3 18.4 18.4.1 18.5 18.5.1 18.5.2 18.5.3
Page 6 of 85
Notes for Analysis 2 (v 1.0)
Anders MunkNielsen
October 2010
1 First lecturevector space over DEFINITION inner procuct on A map; I.e. For all satisfies
(where means complex conjungation,
)
NOTE! Reformulation of If is a vector space over with inner product, then is an inner product space.
EXAMPLE
Let the inner product be
Why this definition? positivity See e.g. and (modulus, length of vector)
Scalar?
NOTE! in this setting, (since we generally have for that )
EXAMPLE infinite dimensional spaces We have
And now we define
Page 7 of 85
Notes for Analysis 2 (v 1.0)
Anders MunkNielsen
October 2010
Lets look at the axioms that must be met
Actually,
1.1
Theorem 1.5 We are assuming that hold. Then the following statements are true
We start with
use assume
with
. consider so this is zero. . (holds for all, so specially or this)
first of all, is trivial.
But we assumed But then EXAMPLE little ell two
(since we think of functions from the natural numbers as series) NOTE is written ell.
DEFINITIONS We now introduce
(another question;
??)
Page 8 of 85
Notes for Analysis 2 (v 1.0)
Anders MunkNielsen
October 2010
1.2
Something is absolutely convergent if and only if is convergent
Suppose , DEFINITION We say, s.t.
THEN IT FOLLOWS that i.e.
Discount inequalities
PROOF first;
second; 1.2.1 Proof, something about convergent
Lets assume that
(since convergent)
means that
and
are square summable, i.e. the infinite sum of squared absolutes is
We may now check the axioms
Now lets check whether
Page 9 of 85
Notes for Analysis 2 (v 1.0)
Anders MunkNielsen
October 2010
(fundamental property of taking modules that
1.3Let
Normsbe an inner product space. Norm, , since the inner product is never negative
No worry about EXAMPLE Let 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): (iii): Page 10 of 85 by assumption on the inner product
Notes for Analysis 2 (v 1.0) FACT PROOF
Anders MunkNielsen
October 2010
(we take squares so we dont have to worry about the square root)
now we use the fact that for
we have
,
1.4
Theorem 1.9: CauchySchwartzand
PROOF Assume and not lin.dep. , show
which is strictly positive for all values of . We need a trick o We need to get rid of for think e.g. of polar decomposition , where has this structure;
Consider now, the special case where
Note now, that then so the equation becomes
since
by creation
This is a quadratic equation in WE KNOW THAT ITS POSITIVE! This means that the discriminant must be negative
Page 11 of 85
Notes for Analysis 2 (v 1.0)
Anders MunkNielsen
October 2010
2 Chapter 1 continued2.1 Recapinner product on for all Norm Defines We proved that And CauchySchwarz Also, Angles between vectors Then we can define associates ok since if
vector space over
so that NOTE! Since we take of , we cant 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 wouldnt know what the angle was AHA this is the price for working with complex numbers.
2.2
Theorem 1.11: Triangular inequality
PROOF Surprisingly nontrivial to prove. We will start by squaring (which we showed last time) since . Now use, if then we can write and then and Page 12 of 85 and
Notes for Analysis 2 (v 1.0) this means that
Anders MunkNielsen which we will use as
October 2010
where we used CauchySchwarz in the last step, ERGO
2.3
Metric
We have to use some axioms for this 1. 2. 3. 2.3.1 Proof of the triangle bandit (3) and
to be a metric
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
) Page 13 of 85
Notes for Analysis 2 (v 1.0) Now we sum them
Anders MunkNielsen
October 2010
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 Lets 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 spacesvector spaceover or Let denote the field and then lets look at . over the field (Danish: field = legeme) But mostly well be thinking of DEFINITION A norm on is a function
Page 14 of 85
Notes for Analysis 2 (v 1.0)
Anders MunkNielsen
October 2010
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
3.1Let
Examplebe a compact metric space, e.g. Consider continuous functions We could define
NOTE
will hold since
continuous on the compact space will have a min/max )
( then the supremum theorem states that CLAIM NOTE Take then Why? on
PROOF of the triangle equality
Now we can say
NICE2KNOW as an exercise, we will show that there is no inner procduct on that norm (unless consists of only one point) 
so that
comes from
IDEA for the proof doesnt satisfy the parallelogram identity which any norm induced by an inner product must (this is used in the proof of the parallelogram identity). Page 15 of 85
Notes for Analysis 2 (v 1.0)
Anders MunkNielsen
October 2010
3.2
ContinuityIf we have two metric spaces, and , and we have a function is an open subset of . , then we say that
RECALL
 i.e. for all open subsets of , the originalmngde by THEOREM (more useful way of thinking about it) 3.2.1
Theorem 2.5: Continuity of addition and scalar multiplication normed vector space the addition is continuous scalar multiplication is continuous
i) ii)
Actually, ii is more difficult than i PROOF OF ii Show that CLARIFYING What does it mean that It means that Or that and and then
Lets look at the animal and make some tricks
and well prove that this becomes a small number for
large
BUT our assumption is that PROBLEM we only know that SOLUTION o o The map means that
and but not that
is continuous
o THEN
Page 16 of 85
Notes for Analysis 2 (v 1.0)
Anders MunkNielsen
October 2010
3.3Let
(Linear) subspacesbe a normed vector space over
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 Closed linear subspace DEFINITION , is a closed linear subspace if and is a subspace in , and , then
3.4
Theorem 2.13normed vector space. Then all finite dimensional linear subspaces are automatically closed.
3.4.1
Examples
what are subspaces? , o i.e. is not closed (in fact, , although )  PROPOSITION PROOF  Consider , ,

CLAIM (hence not closed) We need to look at the norms Page 17 of 85
Notes for Analysis 2 (v 1.0)
Anders MunkNielsen
October 2010
(
is not closed)
4 Lecture 34.1 Recapover (or ) with a norm cant come from any inner product EX inner product space then also normed with doesnt fulfill parallelogram identity)
A normed space is a vector space
BUT doesnt necessarily go the other way, e.g. (proof @
EX Continuous functions gives a metric Then we can define Linear subspace linear subspace (or just subspace) if Closure closed subspace if the closure of is closed ( ) and is a subspace. and let a subset of this be a basis for , call it , ok by theorem 2.13. , where since of only EXAMPLE This shows that In fact Page 18 of 85 and . But then , and is not closed . . But then since otherwise we couldnt have , and this means that for . But , and implies on .
Theorem (an exercise) All finite dimensional subspaces of a normed space are closed. Proof (illustration) use a basis for . Lets consider convergence by with
this means that all the coordinates
can be written as a lincomb
We saw last time CAN BE SHOWN
Notes for Analysis 2 (v 1.0) EXAMPLE , with
Anders MunkNielsen
October 2010
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) is not closed Proof @ contradiction Consider , which is clearly Consider Here, but More rigourusly and note that
Proposition: 
(where weve used; EXAMPLE Here, is not closed ,
, for
( L2 , uniformly THEN
CLAIM Hence PROOF
Now take 10.1.2
and youre done.
L2 almost implies pointwise
THEOREM (MI) CLAIM If And in , then
IDEA it doesnt work for the sequence it self, but the sequence has a subsequence for which it works. 10.1.3 Example A ,
Page 43 of 85
Notes for Analysis 2 (v 1.0)
Anders MunkNielsen
October 2010
pointwise , so , uniformly in
10.1.4
Example B
Write One can show that Thus, Fact: Fact: so , , then we will have convergence in (towards 0). uniformly
BUT it is also clear, that if we just choose
11 Fourier seriesRecall series = rkker = 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 LelenptzConsider
Page 44 of 85
Notes for Analysis 2 (v 1.0) DEFINITION , let
Anders MunkNielsen
October 2010
FACT PROOF FACT PROOF is continuous and hence measurable. since is an orthonormal system ,
(find a antiderivative @ cos/sin) We have used that In fact we have,
WE WANT to prove, that
is a basis (and that its complete)
11.2 Reminders from AN1Fourier coefficients Bessels inequality (AN1 + ch. 4)
(Later, we want to prove that there holds equality) FACT QUESTION Is (anser; yes) Rephrasing; (since its not trivial what the = means) to clarify Put
Q: Does
?
 NOTE if there is convergence If yes, what kind of convergence (uni, point, L2)? Page 45 of 85
Notes for Analysis 2 (v 1.0) A: yes, we have convergence
Anders MunkNielsen
October 2010
11.3 Results from AN1THEOREM A (stning 3.2) If Moreover, if is continuous and (i.e. periodic), then and pointwise. , then is piecewise cts and
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 Then THOEREM C is cts and piecewise uniformly. (e.g. ) and
as in thm. B. Then
(equality in Bessels bandit) 11.3.1 A remark on where your functions live CLAIM (intuition there is one and only one way to expand a periodic function to the entire real line) where , so that and are actually the same. PROOF create a bijective mapping between them.
11.4 The new stuff in chapter 5THEOREM 5.1 is an orthonormal basis for .  (we already know that its an orthonormal set the new thing is that its complete) Hence, the following holds
Page 46 of 85
Notes for Analysis 2 (v 1.0)
Anders MunkNielsen
October 2010
(complete = you cant add another vector to the set so that its still an orthonormal set) ( means that is a nullset (e.g. differ only at finitely many points)
11.4.1
Idea about how the proof goes
Theorem 5.5 (Fejr) If is cts and put

(the average of the first uniformly.
partial sums)
Then 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 is not fulfilled.
piecewise One can show,
We are going to use thm. 5.5 to prove that 5.1 must follow (that TRICK We will be importing the following result THEOREM (MI) (hence, COROLLARY and is dense in wrt. such that .
is a basis)
or equivalently, there is a sequence of functions in is also dense in
that converges to .) for (but so that continuity is .
The idea is, we are almost home by using the function, , defined so that so that its the linear segment that goes to the endpoint maintained). Then the function is still cts. and it converges to our target for
. However, instead of just changing the endpoint, we change the interval
Page 47 of 85
Notes for Analysis 2 (v 1.0)
Anders MunkNielsen
October 2010
12 Fourier12.1 Recapping
THEOREM
is a Hilbert space.
EASY FACT i.e.
is an orthonormal set in .
 TODAY prove that it is in fact a basis Fourier coefficients;

(since
)
are the Fourier coefficients for . Fourier series for

Why is the series in
? insuring the required by theorem converges if .
Since Bessels inequality gives that (something) in the book (which says that
12.2 Theorem 5.1 e_n ON basisTHEOREM 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) Page 48 of 85
Notes for Analysis 2 (v 1.0)
Anders MunkNielsen
October 2010
(i.e. convergence by two norm)
12.3 Theorem 5.5 (Fejr) is continuous and periodic.
Now define

It takes the averages of the fourier coefficients By comparison, in you apply weights of zero to all larger coefficients. . , then ! uniformly as . , i.e. weight 1 to all coefficients up until and
THEOREM uniformly as RECALL THEOREM (AN1) coninuous, REMARKS 
periodic and piecewise
NEW The new theorem doesnt assume piecewise
NAJS2KNOW THEOREM (MI) i.e. COOL think of this as BUT definition of proven. the definition of as set of measurable, finitely square integrable functions makes it a theorem to be is dense in wrt
12.3.1
Recalling metic spaces metric space, , iff , ? , such that . . . .
Consider
What does it mean that there exists ALSO Corrollary is dense (tt) in
is also dense in
.
12.4 Proving the grinerBOOK proves is ONB, then says that uniformlly Page 49 of 85
Notes for Analysis 2 (v 1.0) HERE prove  that is, if PROOF we show this will imply that So we have to show that every Enough to show , , then
Anders MunkNielsen
October 2010 is ONB.
uniformly, then say that (with extras) it follows that uniformly then is dense in by definition of density. in can be approximated with by an s.t. . is an ONB for
CLAIM COR + THM 5.5 THM 5.1
in our clin.
Now the following corollary comes in handy
HOW? For any given we can approximate with a function from We can now apply thm. 5.5; NOW  we proved last time that uniform convergence implies convergence in twonorm 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 its first Fourier coefficients, which might take less resources. , then , hence
sequence in such that determines a function.
Page 50 of 85
Notes for Analysis 2 (v 1.0)
Anders MunkNielsen
October 2010
12.5 Calculating Fourier coefficientsEXAMPLE
Hence,
and we see that the Fourier series for
is finite and given above.
since Quite easy EXAMPLE
.
Fourier coefficients
(use integration by parts) Lets try to apply Parsevals
And the other side
PARSEVAL now gives
EXAMPLE
Page 51 of 85
Notes for Analysis 2 (v 1.0) Take
Anders MunkNielsen
October 2010
will hold, but which one?
(nice function)
Then L Then L EXAMPLE L
What about the derivative? (since
12.6 Proving thm 5.5 is continuous and periodic.
THEOREM PROOF 12.6.1
uniformly as
.
Lemma 0: preeesenting the Fejr Kernel
Put
. Then
where
PROOF
(the integral
is called a convolution (da: foldning))
is called the Fejr kernel. Page 52 of 85
Notes for Analysis 2 (v 1.0) Consider at a point,
Anders MunkNielsen
October 2010
We start by looking at a part of the sum in
gather the bandits
and since
is just constant wrt x
Now, were ready to take some sums
Then
12.6.2 Lemma 5.2 , then
CLAIM
PROOF Tedious calculations. REMARKS 12.6.3 Lemma 5.3
Page 53 of 85
Notes for Analysis 2 (v 1.0)
Anders MunkNielsen
October 2010
REMARKS Ad Note that this means that collapses like a distribution of sorts For no matter , all the area under the graph will end up being in the interval And do note that the total area is constantly so PROOF Clear, since each Difficult to see that BUT easy to see that BUT from the continuity we see that but it isnt defined as such for is continuous and then it must also be at is periodic and is made up of these. extremely fast for .

Here,
only contributes when
.
13 Fourier continued13.1 RecappingStill considering , , Theorem 5.1 in particular, this implies I.e. Page 54 of 85 , , orthonormal basis Fourier coefficients for . is an orthonormal basis for
Notes for Analysis 2 (v 1.0)
Anders MunkNielsen
October 2010
(If seen in
, then we have
, since
consists of kvivalensklasser, i.e. = almost everywhere)
or equivalently
or
(since DEFINE
)
Theorem 5.5 (Fejr) 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 continuous and periodic. Then
LEMMA 0 , Then ,
where
The proof now reduces to examining the properties of the Fejr Kernel LEMMA 5.2 , , then
NOW Page 55 of 85
Notes for Analysis 2 (v 1.0)
Anders MunkNielsen
October 2010
Now we have two different ways of writing 13.1.2 LEMMA 5.3
is easy to see from the sin() expression is most easy shows from the doublesum expression has the interpretation that all the area under the graph of , will end up being in the interval PROOF of Take . for any . , , which is
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. Page 56 of 85
Notes for Analysis 2 (v 1.0) CLAIM PROOF
Anders MunkNielsen
October 2010
(using substitution NOW note that
and
and
)
Now use that
periodic, then
for any
.
Now we can insert
Thus,
Now, again use
periodicity and integrate over another interval of same length
Now put 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 , , , Page 57 of 85 .
First, we use that since
Notes for Analysis 2 (v 1.0)
Anders MunkNielsen
October 2010
FIRST the
NOW remember how Thus for all
was chosen! is also in our interval since since we consider . periodic functions) which is equiv to we have that )
(reminder: Hence (dont get confused that now,
NOW
Here, uset hat
o
Here, perform substitution with
,
. Also
,
(since we could choose NOW (but it is similar) This concludes the proof.
such that
.)
13.2 LemmaHilber space with orthonormal basis . Let . Then
(which resembles the formula PROOF
, just set
)
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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 CauchySchwarz)
13.3 Thm 5.6 + cor 5.7
Suppose Then let THEN
or alternatively or alternatively
, where ,
AND
ALTERNATIVELY We have shown that (where Where means isomorphic to, i.e. for all practical bandits theyre the same)
WHY? Since and
AND the theorem above shows us the connection between the function and the sequence of Fourier coefficients. Page 59 of 85
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13.4 Theorem 5.8Given there exists a polynomial such that i.e. all continuous functions can be approximated by polynomials. PROOF (flavor of it) Lets scale so that Here, we have that THE TASK approximate BEGIN The definition is and and uniformly with a polynomial! with (no big deal, just a scaling) , then for all and for all , , ,
Enough to show that HURRAY! We now that
can be approximated by polynomials (the rest is simply linear combinations) can be approximated by its 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 (or or ) A linear functional on EXAMPLE Then consider ( is a Hilber space) EXAMPLE (set of all touples) If EXAMPLE Page 60 of 85 , consider given by is a linear map
Notes for Analysis 2 (v 1.0) Hilbert space, , consider
Anders MunkNielsen linear functional by
October 2010
(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 spacesMain object of interest in this chapter THEOREM 6.8 but also 6.3
14.1 St i gnagDEFINITION Let vector space over . A linear functional [liner functional] is a linear map NOTE EXAMPLE , so (and is linear) EXAMPLE Hilbert space, This is linear since the inner product is linear in the first variable (and EXAMPLE If we want EXAMPLE , If EXAMPLE then is a linear functional Page 61 of 85 , then must be is fixed) always
(a function but generalized to come from any vector space )
Notes for Analysis 2 (v 1.0)
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is a linear functional.
14.2 Theorem 6.3POINT 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 linear functional E The following are equivalent (the book only has 3, we write 4).
where REMARKS PROOF CLAIM Assume Note, always true that subspace is also closed.
. is a linear subspace of by linearity of . The new thing is that this
is Exercise 6.8 (difficult!) is trivial (if continuous everywhere, then in particular continuous at continuous at . . )
We will use a different definition, namely the  definition of continuoity at True for CLAIM (the claim implies that PROOF Take Then We also have But now we have CLAIM Assume CLAIM PROOF . by with . Put . . . Hence .
which would finish our proof)
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Notes for Analysis 2 (v 1.0) We know, that if Now take Put Then Also, By Show continuous. Note . since and arbitrary but (ok since
Anders MunkNielsen then (if ) . (by definition of
October 2010 ). )
the original claim is already proved since
.
by the claim we just proved. Take a sequence But since 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 , in by definition. , we now have that which is the def of cont.!

NOTE we dont assume the existence of an inverted function. is said to continuous or bounded if condition in theorem 6.3 hold. (i.e. bounded = continuous)
NOTE A linear functional on a normed space 
14.3 Norm of a bounded linear functionalDEFINITION If THEOREM (=claim) (where the previous DEFINITION plays the role of in the proof from earlier so were just giving it a new name) is a bounded linear functional on , then put
normed vector space.
VERY important thing! Here, we will only be interested in the dual space of a Hilbert space. Page 63 of 85
Notes for Analysis 2 (v 1.0) THOEREM is a Banach space (
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)
the book also proves that the norm on this space is in fact a norm. 14.3.1 Examples
Hilber space, (where the inequality is by CauchySchwarz, my homies) Hence CLAIM PROOF We already have Assume Put Then EXAMPLE , Is 1. 2. ( Lets look 1. This shows It is also true that (consider . , ). , . bounded in ? , so that since . is a particular vector of length 1 and is sup over such vectors is bounded (continuous)
IMPORTANT COMMENT He just said that is a Banach space but spaces where is not! (perhaps) it is the converse (??!?!?!?!) Recall the difference from measurability and
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14.4 CombojoeFACT Hilbert space, closed subspace, , then RECALL Why the fact? , hence Another way, Pick , then and then Otherwise , then . . for and .
14.5 Climax: Thoerem 6.8 (RieszFrecht)Let be a Hilbert space. The following holds Let where This means that every bounded linear functional on and Hence, the map , satisfies are of the inner product form be a bounded linear functional on
meaning that we can think of it as but unfortunately . REMARKS Weve already proved PROOF is already proved CLAIM PROOF holds
is only conjugated linear and not linear.
First we need theorem 1.5(iv): , (easy proof, follows from 1.5(iv)) CLAIM PROOF Take CASE 1 holds bounded linear functional
so that
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Notes for Analysis 2 (v 1.0) , now take CASE 2 . Then Since o , Put Then , (note . .
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is a closed linear subspace of . . Then so that where .
Now we have found a vector of length 1 in the orthogonal complement to Now take consider , and note
.
is just a number.
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) Easy to see Bounded? linear functional on . for some
Well, we can see that
 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 spacesTwo main reasons for studying Hilbert spaces; Fourier and Operators. Operators on Hilbert spaces; motivated as a correct mathematical formulation of Quantum Mechanics
15.1 Lets govector spaces over . A mapping, is linear if , i.e. we almost view it as a product of for some
NOTE we often write EXAMPLE ,
and . matrix
is linear if and only if
Thus,
15.2 Thm 7.4Suppose 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 Page 67 of 85
Notes for Analysis 2 (v 1.0) If is bounded, then
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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 its input.
15.3 Continuity of linear mapsAre all continuous? YES! Because is finitely dimensional  (can be proven to hold for any finitely dimensional vector space) Onwards Given ,  No simple formula One (stupid) estimate: matrix. What is ? (since is finite)
15.4 Various examplesEXAMPLE (i) Let
Here, CLAIM PROOF Hence, This gives . EXAMPLE (ii), the Fibonacci map Now, Tak arbitrary, First, . Hence .
By
(the stupid estimate),
CLAIM REMARK
the golden value
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Notes for Analysis 2 (v 1.0) Eigenvalues of ; There are eigenvectors, AND such that 
Anders MunkNielsen , , such that is an orthonormal basis for . . and
October 2010
EXTRA REMARK .
The reason is that is symmetric (or rather, somethingsomething complex symmetric, but that is just like being symmetric when all elements are real) PROOF (incomplete similar to the previous) take . Since is a basis, we can write any such vector as .
THE RESULT the maximum eigenvalue must be the norm of EXAMPLE (iii)
gives that Eigenvalues of are (note that and is upper triangular) . is not equal to the All eigenvectors are
NOTE then the eigenvectors do not represent an orthonormal basis and hence largest eigenvalue.
15.5 An operator interpretable as an infinitely dimensional matrix, , means that Now we want to define CLAIM PROOF LINEAR One must show . . Both sides are functions, so we evaluate at arbitrary , (pointwise multiplication, linear, bounded and . ) linear. , hence, . We want it to satisfy such that is measurable and .
Clearly measurable (?!) But finite integral? Page 69 of 85
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Hence, Hence NOTE linear and BECAUSE EXAMPLE , Hence, Show i.e. find . .
.
st . . Note that .
then
,
,
. , and
this turns out to be impossible! But a little less might do if we can only find since then we would just take the Take one can now check that Now see that . and get what we want (sup = 2 2).
But this proves
so that
15.6 Example integral operatorsRecall the Fejr kernel,
PURPOSE NOW  Input = , output  This is actually a linear map. More generally, Consider the two intervals Consider the function And Page 70 of 85 and in .
Notes for Analysis 2 (v 1.0) And the function
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at a given value, is

Fejr 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 operatorsEXAMPLE Define the operator We want to write . contains many functions which are not differentiable. . ) . . . (the book write )
However, this is not generally possible since and even if it is, its not sure that domain of (actually dense) NOTE INSTEAD, let Clearly, BUT,
is linear ( is unbounded;
CLAIM is unbounded; PROOF (outline)
Note, (since we get an down when we differentiate)
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15.8 blahEXAMPLE
Again, we must restrict us to the domain of ,
Then Consider
,
is a dense subspace.
We can easily see
Hence, Aha, so for all , is an eigenvalue for with eigenvector . .
Moreover, we see that the set of eigenvectors, NOW Why does this show that It now follows that Finally, is unbounded?
, is an orthonormal basis for
Wow, have we now defined the derivateive of any well not exactly, it only works when function?
But this allows us to redefine the set .
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16 Chapter 7 contd16.1 SpectrumBanach space (or Hilbert space) (where .) A
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, Ingredients for the proof We showed these last time. PROOF Consider Now define is a continuous mapping. Now we can define the spectrum of PROOF Take 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 . Page 73 of 85 . is closed (omvendte af ben) (Kugle = Ball)
Notes for Analysis 2 (v 1.0) EXAMPLE
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16.2 Adjoint operator
More generally
16.2.1
Theorem (linAlg linear,
Hermitian (selfadjoint) case, THEOREM Hilbert spaces, Then SPECIAL CASE , then it is formulated as MOREOVER LEMMA (not in book) Let . Then
.
.
PROOF of lemma Let ,
put (note that so
and note ) Page 74 of 85
Notes for Analysis 2 (v 1.0) Hence, Since
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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 is linear since CLAIM PROOF is linear and is linear in the first variable. is bounded
Hence, (sicne qed By RieszFrechet Put Then Hence So we must show that CLAIM PROOF Take , (we still have Look at the following ) is linear is linear and that is bounded (and then we have proved existence) such that )
(SO FAR we have proved that a mapping exists we need to prove that its linear and bounded)
We now want to conclude that this implies that This is true because (chapter 1): qed CLAIM PROOF is bounded
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Notes for Analysis 2 (v 1.0) (now use
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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 16.2.2 is an adjoint operator and by the theorem from before, it is the unique.
Thoerem A** = A
THEOREM REMARKS Note that PROOF Take Hence, we have that Page 76 of 85 and . and look at for all and use that . is the adjoint.
, so by thm. from chapter 1
Notes for Analysis 2 (v 1.0) 16.2.3 sets, Then (note that THEOREM PROOF Take . and not ) Sammensatte operatorer
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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 , , then
(i.e. the adjointoperator is conjungated linear)
16.3 Hermitian operatorsDefinition Hilbert space, .
Then is Hermetian (da: hermitesk) or selfadjoint (da: selvadjungeret) 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 . Which of the following are Hermetian?

is is not Page 77 of 85
Notes for Analysis 2 (v 1.0) is not is. EXAMPLE We showed that
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But now we can state the following i.e. that being hermetian is equal to being real. INSHOT (indskud) Take EXAMPLE , put CLAIM PROOF Take and , are hermetian and . . Now use that is conjugated linear . , then , .
Easy to see that
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 (mgwtf?) THEOREM THEOREM EXAMPLE , multiplication operator The thing is We can almost see this fact because Page 78 of 85 , then , hermetian, then
Notes for Analysis 2 (v 1.0)
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so PROOF OF THEOREM Assume Now since , We assume that This is so because for , .
for some
. (proves the theorem for multiplication operators)
INDICATION OF WHY THIS HERE HOLDS
, this means that . Show that , and . if . .
and want to show that
Now by the first part of the proof, But we now that By lemma, this means that
by the assumption and hence the imaginary part of this must be zero. and hence . Since , .
17 On the examIf 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 theres no proof for them (if theyre in a bistning) 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 syllabus18.1 Normed spacesvector space over 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 Page 79 of 85
Notes for Analysis 2 (v 1.0) Not all subspaces are closed, e.g.
Anders MunkNielsen is not closed (i.e. taken in is not equal to )
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EXERCISE Normed space, finite dimensional subspace, then is closed. COMPLETE SPACE BANACH SPACE complete if all Cauchy sequences are convergent. is a Banach space if normed space and complete.
18.2 Inner product spacesINNER PRODUCT SPACES vector space over with inner product 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 , In In Hilbert space with norm Not Hilbert space I.e. there are Cauchy sequences in Contains but also piecewise functions and even more exotic functions. is the completion of with respect to . , include it in and if their limit is not in that arent convergent. , , Hilbert space with (and this will converge since for a .
EXAMPLES of inner product spaces In

Intuitive understanding as well.
i.e. take all Cauchy sequences in Sort of like is the completion of .
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Notes for Analysis 2 (v 1.0) 18.2.1 Hilbert space
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I.e. all functions in
is the limit of a sequence of functions in
Orthonormal sets
Relevant in Hilbert spaces. ORTHONORMAL SET Let Then be a Hilbert space. Let is an orthonormal set if , . . I.e. and .
FINITE CASE INFINITE CASE EXAMPLE , here the orthonormal basis is or (instead of using an arbitrary index set, , we will be using the natural numbers)
,
Since we may write as Orthonormal system (even ON basis)
.
18.2.2 Basis
, is an orthonormal set (even ON basis)
COMPLETE An orthonormal set is complete if i.e. is the maximal wrt being an ON set is an orthonormal basis for if it is an ON set and is complete ORTHONORMAL BASIS THEOREM Let be a Hilbert space, and let ON
be an orthonormal set. Then the following are equivalent
P Page 81 of 85
Notes for Analysis 2 (v 1.0) 18.2.3 By we mean that
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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, Put Let orthonormal set in to . is given by (which is also the closed linear span(!)) . Then the closest point, , in
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 (didnt quite hear that)
18.3 Fourier series
THEOREM 5.1 (with
is an orthonormal basis for .
COR ,  NOTE! There may be confusion! COR Book: Lectures: , then
Page 82 of 85
Notes for Analysis 2 (v 1.0) Put We also say
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, then we are saying that
PRACTICAL REMARK  Finding coefficients can be tedious since the inner product involves an integral  But for some nicer functions, it will be doable THEOREM Fejr Then for , (or continuous and periodic) we have
18.4 Linear functionalnormed vector space. LINEAR FUNCTIONAL on . Let linear functional. Then we may define OPERATOR NORM NOTE BOUNDED OPERATOR THEOREM If is bounded if
linear functional, then the following are equivalent
18.4.1
Dual space is
DUAL SPACE the dual space to is a Banach space  This holds even if EXAMPLE Hilbert space,
itself is not complete . Then we can define Page 83 of 85
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From CauchySchwarz (more or less) we get Since CS gives bounded linear functional , and compare it with
THEOREM (RieszFrecht) Hilbert space, Then Or more precisely Also, (and )
18.5 Operatorsnormed spaces If linear, we can (in analogy to the functional norm) define the operator norm
BOUNDED THEOREM Notation Relation:
is bounded bounded continuous
Sammensatte operatorer  We get that INVERTIBILITY 18.5.1 invertible if Where Spectrum then we define Properties of the spectrum 18.5.2 closed Generalizes the notion of eigenvalues from linear algebra Adjoint Hilbert spaces, Page 84 of 85 and then we can associate
Notes for Analysis 2 (v 1.0)
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 We used RieszFrecht to prove this PROPERTIES 
18.5.3
Hermitian operators Hilbert space,
Consider the Hilbert space then we associate an operator and by
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