Functional Analysis I

Question 1

Define dense and nowhere dense

Answer 1

pg. 6, Beispiel 1.2.1 vi)

Question 2

Prove Satz: M metric space. U open in M, A=M\U. TFAE:

i) U is open and dense
ii) A is closed and nowhere dense

Answer 2

Satz 1.2.1 , pg. 6

Question 3

Characterize closure in metric space

Answer 3

closure(A)={x limit point of xn in A}

Question 4

Define Cauchy sequence

Answer 4

(xk)k Cauchy in metric space M if d(xk,xl)–>0.

Question 5

Define completeness

Answer 5

A metric space M is complete if every Cauchy sequence in M converges in M.

Question 6

Define meager, non-meager, residual sets and the Baire categories.
Give an example of a meager set.
What are subsets and/or unions of meager sets?

Answer 6

Def 1.3.1, pg. 9

Beispiel 1.3.2

Question 7

State Baire’s theorem,

Prove for 3+

Answer 7

Satz 1.3.2, pg. 9

Question 8

Can Lebesgue-Nullsets and meager sets be characterized?

Answer 8

No, there is example of either not being the other. For an example of a non-meager nullset give Beispiel 1.3.3, pg 10

Question 9

State the “principle of uniform boundedness”.

For 5. give its proof

Answer 9

Satz 1.4.3, pg 13

Question 10

Define Banach space

Define Hilbert space

Answer 10

A Banach space is a normed space, which is complete w.r.t. the metric induced by the norm.
A Hilbert space is an inner product space, which is also a complete metric space w.r.t. the metric induced by the norm induced by the inner product.

Question 11

What is B(M,X) and when is it complete? Prove it

Answer 11

Bsp. 2.1.1 i), pg. 15

Question 12

Define isometry.

Answer 12

(M,d),(M,d) metric spaces. T:M–>M* is an isometry if d*(T(x),T(y))=d(x,y) for all x,y in M.

Question 13

Define equivalence of norms.

Answer 13

|.|_1 and |.|_2 are equivalent iff there is a constant C>0 s.t. for every x in X: |x|_1 / C =< |x|_2 =< C * |x|_1.

Question 14

Give a sufficient condition for two norms to be equivalent?

5: Prove it

Answer 14

The two norms are defined on a finite dimensional R(/or C)-vector space.
Satz 2.1.2, pg. 18

Question 15

What can we say about finite dimensional subspaces of normed vector spaces?
5: Prove it

Answer 15

They are complete, closed and all norms are equivalent. (satz 2.1.3, pg 18)

Question 16

Give an example of a subspace of a complete subspace, which is neither closed nor Cauchy.

Answer 16

Consider C^0([0,2]) as a subspace of L^1([0,2]) with the L^1 norm. Then (fn)n where fn(t):=t^n for t in [0,1) and else fn(t):=1 is Cauchy in L^1 but its limit is not in the subspace C^0. (beispiel 2.1.3, pg 19)

Question 17

Characterize compact (metric spaces) in metric spaces.

Answer 17

K s.s.o. M is (sequentially) compact if every sequence has a convergent subsequence.

Question 18

Characterize finite dimensional normed vector spaces.
For 5 (one direction): prove

Answer 18

A normed vector space is finite dimensional iff its unit sphere is compact. (satz 2.1.4, pg 19)

Question 19

Characterize linear continuous functions:

Prove: 4+

Answer 19

i) A is continuous in 0 in X.
ii) A is continuous everywhere.
iii) A is uniformly continuous.
iv) A is Lipschitz continuous.
v) A’s operator norm is finite (well-defined).
satz 2.2.1, pg 21

Question 20

What is a sufficient condition for a linear function to be continuous?
Give an example of a linear non-contin function
Prove for 3

Answer 20

X finite-dimensional. Prove Lipschitz contin, Satz 2.2.2, pg. 21
Counter example: X=Y=C^0([0,1]), A=id, where we equip X with L^1 norm and Y with C^0 norm

Question 21

When is L(X,Y) a Banach space?

Proof: 5+

Answer 21

L(X,Y) is a Banach space, if Y is. Satz 2.2.4, pg 22.

Question 22

Define the Spektralradius. Set it in relation to the operator norm.
Prove for 5+

Answer 22

Satz 2.2.6, pg 24

Question 23

What is the Neumann series of an operator and when is it well-defined, what is is equal to then?

Answer 23

Satz 2.2.7, pg 24

Question 24

What can we say about Gl(X) topologically when a certain ?

condition (which?) is fulfilled? Prove for 4

Answer 24

When X is Banach, Gl(X) is open in L(X).

Satz 2.2.8, pg 25

Question 25

Given X,Y can we define a normed space (X/Y,||..||_X/Y)? Is it Banach`?

Answer 25

Satz 2.3.1, pg. 25

Question 26

Define scalar product

Answer 26

def 2.4.1 pg 27

Question 27

Prove Cauchy Schwarz

Answer 27

WLOG |x|=|y|=1. Let t=conj(x,y) and note (x,y-tx)=0. | Now 1=|y|^2=|tx+(y-tx)|^2=|t|^2*|x|^2+|y-tx|^2>=|t|^2=|(x,y)|^2.

Question 28

What can we say about orth(Y)=Y^? Prove it.

Answer 28

``` lemma 2.4.2, pg. 28, And more or less: lemma 2.4.3, pg. 28, Korollar 2.4.1, pg 29, satz 2.4.1, pg. 29: X=Y⊕Y^⊥ + inclusions and Lemma 2.4.4, pg 29 ```

Question 29

State the Banach-Steinhaus Theorem. | Prove for 3.

Answer 29

Satz 3.1.1, pg. 31

Question 30

Show L(X,Y) is closed if X is complete.

Answer 30

Anwendung 3.1.1, pg. 32, (von Banach Steinhaus)

Question 31

Define open operator and state the Open Mapping Theorem.

Answer 31

Def 3.2.1, Satz 3.2.1 pg. 32

Question 32

Define closed operator.

Answer 32

(Not definition from topology!) X,Y normed spaces. D(A) s.s.o. X lin subspace. A:D(A)-->Y linear is a closed if its graph Gamma_A={(x,Ax); x in D(A)} is closed in XxY (for example w.r.t. the norm: |(x,y)|:=|x|+|y|) (Def 3.3.1, pg. 35) (Note if D(A)=X and A in L(X,Y), then A is a closed operator. Pf: (xk,yk), then xk->x => yk=Axk->y=Ax since A contin and (x,y)=(x,Ax) in graph)

Question 33

Show harder direction of Closed Graph Theorem. | Reformulate following bem.

Answer 33

X,Y Banach, A:X-->Y linear. Then: A in L(X,Y) if and only if A is a closed operator. (Note: D(A)=X. Counter example when fails due to this X=C^0([0,1]), A=d/dt on C^1([0,1])) Satz 3.3.1, pg. 35

Question 34

State the Theorem of the Continuous Inverse | Which proof is its proof analogous to?

Answer 34

Satz 3.3.2, pg 37 | anal to satz 3.3.1 closed mapping thm

Question 35

Define linear graph.

Answer 35

Let X,Y be normed vector spaces and Gamma a linear subspace of XxY. Gamma is called a linear graph, if (x,y1),(x,y2) in Gamma implies that y1=y2. Or equivalently if (0,y) in Gamma implies y=0. See Bem. 3.4.1 for the relation between the graph and linear operator.

Question 36

Define extension and closable operator.

Answer 36

Def 3.4.2, pg 38

Question 37

Prove A is closable iff ((xk,yk)) s.s.o. Gamma_A: | If xk-->0 and yk=Axk, then y=0.

Answer 37

A is closable iff clos(Gamma_A) is a linear graph and that is the case iff (0,y) in clos(Gamma_A) implies y=0. Satz 3.4.1, pg 38

Question 38

Prove that a linear and contin operator is closable

Answer 38

(xk,yk=A=xk) in Graph_A s.t. xk-->0. Since A contin, |Axk|=0

Question 39

Is A=d/dt closable? | Prove for 5+

Answer 39

We showed that any differential operator: | A: C^inf_c(Omega) s.s.o. L^p(Omega)-->L^p(Omega) is closable for any p in [1,inf].

Question 40

State the Lemma for the Calculus of Variations ("Fundamentallemma der Variationsrechnung"). For 4: prove it

Answer 40

Satz 3.4.3, pg. 40

Question 41

Define sublinear.

Answer 41

X R-VS. p:X-->R sublinear if 1) p(a*x)=a*p(x) for all x in X and a \geq 0, 2) p(x+y) \leq p(x) + p(y) Def 4.1.1, pg 43

Question 42

State Hellinger-Töplitz | Prove for 4+

Answer 42

Beispiel 3.3.2, pg. 36

Question 43

State the Hahn-Banach thm. | Sketch its proof for 3+

Answer 43

Satz 4.1.1, pg 43

Question 44

Define C-sublinear | Additional property that R-sublin doesnt have? (that follows)

Answer 44

def. 4.1.2, pg 45, p>=0

Question 45

Reformulate Hahn-Banach for C-vector spaces

Answer 45

Satz 4.1.2, pg 45

Question 46

State and prove the theorem of "dominated extension" ("Dominierte Fortsetzung")

Answer 46

Satz 4.1.3, pg 46

Question 47

Define dualspace

Answer 47

(X,||.||) normed VS. X*:=L(X,R) dualspace of X.

Question 48

For every x in X there is an x* in X*, s.t. ??? | Prove for 3+

Answer 48

s.t. x*(x)==||x*||^2=||x||^2 (satz 4.2.1, pg 46)

Question 49

For all x in X: sup{ |(x*,x)| : ||x*|| =< 1}=? | For all x* in X*: sup{ |(x*,x)| : ||x|| =< 1}=?

Answer 49

=||x|| =||x*|| i.e. takes the supremum Satz 4.2.2, pg 47

Question 50

Let x != y in X then ? | Prove for 2+

Answer 50

there is an l in X* s.t. l(x) != l(y) | Satz 4.2.3, pg 47

Question 51

Satz: Let M s.s.o X be a closed linear subspace s.t. M != X, and let x0 in X\M with d=dist(x0,M):=...? Then...? Prove for 5+

Answer 51

Satz 4.2.4, pg 47

Question 52

Define annihilator

Answer 52

def 4.2.2, page 48

Question 53

Define j_y and J | What properties do they have?

Answer 53

pg 49, satz 4.3.1

Question 54

State Riesz representation theorem (Rieszscher Darstellungssatz) State strategy of proof for 5+

Answer 54

satz 4.3.2, pg 49

Question 55

State the Lax-Milgram theorem, | its corollary for 5+

Answer 55

satz 4.3.3, pg 51

Question 56

What can we say about the dualspace of L^p under certain (which?) conditions of the measure space?

Answer 56

satz 4.4.1, lemma 4.4.1 and lemma 4.4.2 pg 53

Question 57

State the separation theorem for convex sets

Answer 57

satz 4.5.1, pg 56

Question 58

Define the Minkowski-Funktional

Answer 58

pg 57

Question 59

define extremal subsets and points | give two contrasting examples in R^2

Answer 59

def 4.5.1, pg 58

Question 60

What can we say about subsets of extremal subsets? Prove for 3+ State a set which is extremal in every compact set PRove for 4+

Answer 60

Lemma 4.5.1, pg 59 | Lemma 4.5.2, pg 59

Question 61

State the Krein-Milman theorem (only pf for ii) for 5+)

Answer 61

satz 4.5.2 pg 59 and satz 4.5.3 pg 60

Question 62

Define convex hull

Answer 62

def 4.5.2, pg 60

Question 63

Define weakly convergent

Answer 63

definition 4.6.1, pg 60

Question 64

If (xk)k converges weakly, what can we say about (xk)k? Prove for 5+ Hint: liminf |xk|?

Answer 64

satz 4.6.1, pg 61

Question 65

Define weak topology How do the closed sets of the topology and the weak topology relate? How do weak convergence and the weak topology relate?

Answer 65

definition 4.6.2, pg 61 | lemma 4.6.1, pg 62

Question 66

Define weakly sequentially closed

Answer 66

definition 4.6.3, pg 62

Question 67

How do weakly closed, weakly sequentially closed and closed relate? Prove for 4+

Answer 67

weakly closed => w.s.c => closed | Lemma 4.6.2, pg 62

Question 68

Give a counter example that disproves that closed does not imply weakly sequentially closed

Answer 68

the 1-sphere in l^2 is closed but according to beispiel 4.6.1 is not weakly sequentially closed (beispiel 4.6.2, pg 62)

Question 69

What can we say about a convex set and the weak topology? | Pf for 4+

Answer 69

the weak closure is equal to the closure (satz 4.6.2, pg 62)

Question 70

State Mazur's lemma

Answer 70

satz 4.6.3, pg 63

Question 71

When are the weak and strong topology equivalent? | What do the sets Omega_{l,U} (i.e. the basis sets of the weak topology) contain when the condition is not satisfied?

Answer 71

when dimX

Question 72

Define bidualspace

Answer 72

``` X**=(X*)*=L(X*,R) is the bidualspace of X. def 5.1.1., pg 65 ```

Question 73

Define the canonical embedding of the dual space of a space into the space state the properties of this embedding

Answer 73

def 5.1.1., pg 65

Question 74

``` Define reflexivity of a normed space State some examples and non examples 1 ex for 3, 3 ex for 4, 4 ex for 5 ```

Answer 74

definition 5.1.2, pg 65 | Another non example L^inf (beispiel 5.1.2, pg 67)

Question 75

If X is reflexiv then X is also... | Proof for 3+

Answer 75

complete (bemerkung 5.1.1, pg 66), reflexive (Satz 5.1.2.)

Question 76

Relate X and X* by reflexivity.

Answer 76

Satz 5.1.2, pg 66 i) X* ii) complete, reflexiv

Question 77

Under what sufficient conditions is a subspace of a normed space reflexiv?

Answer 77

X is reflexiv, Y closed subspace of Y, then Y is also reflexiv. Satz 5.1.3, pg 67

Question 78

Define separability (+characterization)

Answer 78

Note M is metric space: definition 5.2.1/bemerkung 5.2.1, pg 68

Question 79

Under which conditions are subsets of _____ spaces are separable? Prove

Answer 79

Subsets of separable metric spaces are separable w.r.t. the induced metric Satz 5.2.1. pg 68

Question 80

Relate dual spaces and separability | Prove 1 for 3, both for 4

Answer 80

Satz 5.2.2 (pg 68): X normed R vector space: i) X* separable, then X is separable ii) X separable and reflexiv, then X* is also separable

Question 81

Define weakly convergent*

Answer 81

definition 5.3.1, pg 69

Question 82

State the Banach-Alaoglu theorem | Proof for 4+

Answer 82

satz 5.3.1, pg 70

Question 83

State the Eberlein-ˇSmulyan theorem | Prove for 3+

Answer 83

satz 5.3.2, pg 72

Question 84

State the approximation theorem | Prove for 3+

Answer 84

Satz 5.3.3, pg 72

Question 85

Define w.s.l.s.c. (weakly sequentially lower semi-continuous) "schwach folgen-unterhalb-stetig"

Answer 85

definition 5.4.1, pg 73

Question 86

Define coercive

Answer 86

def 5.4.2, pg 74

Question 87

State the variation principle (“Variationsprinzip”) | Prove for 3+

Answer 87

satz 5.4.1, page 74

Question 88

Define dual operator

Answer 88

definition 6.1.1., pg 77

Question 89

Relate an operator and its dual with an equation | Prove for 5+

Answer 89

||A||=||A*|| | Satz 6.1.1, page 77

Question 90

What holds for densely defined dual operator's dual and "any" "extensions"? Pf for 5+

Answer 90

Satz: If A:D_A s.s.o. X --> Y densely defined. Then: i) A*:D_A* s.s.o. Y* --> X* is closed ii) A "s.s.o." B then B* "s.s.o." A* Satz 6.1.2, page 78

Question 91

State Banach's closed range theorem

Answer 91

Satz 6.2.1, pg 79 | Wonder if I should add the next Satz and lemma?

Question 92

Define compact operator

Answer 92

Definition 6.2.1, pg 80

Question 93

What holds for cpt operators and weakly convergent sequences? Pf for 5?

Answer 93

T cpt in L(X) and xk-w->x in X, then Txk-->Tx (i.e. converges strongly in X) Lem 6.2.2, pg 80

Question 94

State Arzéla-Ascoli

Answer 94

satz 6.3.1, pg 81

Question 95

State the Fréchet-Kolmogorov theorem (if manageable)

Answer 95

satz 6.3.2, pg 82

Question 96

Define adjoint operator

Answer 96

definition 6.4.1, pg 84

Question 97

How to the dual operator and adjoint operator relate?

Answer 97

Bemerkung 6.4.1, pg 85

Question 98

Define symmetric and self adjoint

Answer 98

definition 6.4.2, pg 85

Question 99

Define resolvent set and spectrum

Answer 99

definition 6.5.1, pg 86

Question 100

Define resolvent and describe how the resolvent and resolvent set relate. And give the topological properties of resolvent set and spectrum that it implies.

Answer 100

definition 6.5.2, pg 87 | satz 6.5.1

Question 101

Do two resolvents commute (w.r.t. the same operator)?

Answer 101

Yes (satz 6.5.2, pg 87)

Question 102

Define eigenvalue, eigenspace, point spectrum, continuous spectrum and restspectrum

Answer 102

definition 6.5.3, pg 88

Question 103

How do the spectral radius, the spectrum and the resolvent set relate?

Answer 103

Satz 6.5.3, pg 89

Question 104

Define umrundet, i.e. surrounds and "schar"

Answer 104

definition 6.5.4, pg 90

Question 105

State the spectral (mapping) theorem, Pf for 4+, (Lemma 6.5.2 may be used)

Answer 105

Satz 6.5.4, pg 91

Question 106

What hold for the point spectrum when A is symmetric? | Pf for 4+

Answer 106

The eigenvalues are real, i.e. the point spectrum is a subset of R (satz 6.6.1, pg 92)

Question 107

Define weak derivative

Answer 107

definition 6.6.1, pg 92

Question 108

Give characterizations of selfadjointness

Answer 108

Satz 6.6.2, pg 95

Question 109

Define normal and unitary

Answer 109

definition 6.7.1, pg 96

Question 110

What follows for the spectral radius of a normal operator on a Hilbert space?

Answer 110

satz 6.7.1, pg 98

Question 111

State Baire's theorem on functions (long proof)

Answer 111

satz 1.4.1, pg 10

Question 112

Prove that for K cpt in X, l in X*, lambda=min{ l(x) : x i n K}. Then K_lambda = {x in K : l(x)=lambda} is extremal in K

Answer 112

lemma 4.5.2, pg 59

Question 113

Prove: a:H^2-->R bilin, cont. and f in H*. Then there is precisely one x in H s.t. a(x,y)=f(y) for all y in H and |x|=< || f ||/lambda.

Answer 113

korollar 4.3.1, page 52

Question 114

Define and characterize topologically complemented subspaces V s.s.o. X. Prove why they are closed.

Answer 114

Exercise 3, sheet 4