Functional Analysis I Flashcards
Define dense and nowhere dense
pg. 6, Beispiel 1.2.1 vi)
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
Satz 1.2.1 , pg. 6
Characterize closure in metric space
closure(A)={x limit point of xn in A}
Define Cauchy sequence
(xk)k Cauchy in metric space M if d(xk,xl)–>0.
Define completeness
A metric space M is complete if every Cauchy sequence in M converges in M.
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?
Def 1.3.1, pg. 9
Beispiel 1.3.2
State Baire’s theorem,
Prove for 3+
Satz 1.3.2, pg. 9
Can Lebesgue-Nullsets and meager sets be characterized?
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
State the “principle of uniform boundedness”.
For 5. give its proof
Satz 1.4.3, pg 13
Define Banach space
Define Hilbert space
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.
What is B(M,X) and when is it complete? Prove it
Bsp. 2.1.1 i), pg. 15
Define isometry.
(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.
Define equivalence of norms.
|.|_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.
Give a sufficient condition for two norms to be equivalent?
5: Prove it
The two norms are defined on a finite dimensional R(/or C)-vector space.
Satz 2.1.2, pg. 18
What can we say about finite dimensional subspaces of normed vector spaces?
5: Prove it
They are complete, closed and all norms are equivalent. (satz 2.1.3, pg 18)
Give an example of a subspace of a complete subspace, which is neither closed nor Cauchy.
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)
Characterize compact (metric spaces) in metric spaces.
K s.s.o. M is (sequentially) compact if every sequence has a convergent subsequence.
Characterize finite dimensional normed vector spaces. For 5 (one direction): prove
A normed vector space is finite dimensional iff its unit sphere is compact. (satz 2.1.4, pg 19)
Characterize linear continuous functions:
Prove: 4+
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
What is a sufficient condition for a linear function to be continuous?
Give an example of a linear non-contin function
Prove for 3
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
When is L(X,Y) a Banach space?
Proof: 5+
L(X,Y) is a Banach space, if Y is. Satz 2.2.4, pg 22.
Define the Spektralradius. Set it in relation to the operator norm.
Prove for 5+
Satz 2.2.6, pg 24
What is the Neumann series of an operator and when is it well-defined, what is is equal to then?
Satz 2.2.7, pg 24
What can we say about Gl(X) topologically when a certain ?
condition (which?) is fulfilled? Prove for 4
When X is Banach, Gl(X) is open in L(X).
Satz 2.2.8, pg 25
Given X,Y can we define a normed space (X/Y,||..||_X/Y)? Is it Banach`?
Satz 2.3.1, pg. 25
Define scalar product
def 2.4.1 pg 27
Prove Cauchy Schwarz
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.
What can we say about orth(Y)=Y^? Prove it.
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
State the Banach-Steinhaus Theorem.
Prove for 3.
Satz 3.1.1, pg. 31
Show L(X,Y) is closed if X is complete.
Anwendung 3.1.1, pg. 32, (von Banach Steinhaus)
Define open operator and state the Open Mapping Theorem.
Def 3.2.1, Satz 3.2.1 pg. 32
Define closed operator.
(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)
Show harder direction of Closed Graph Theorem.
Reformulate following bem.
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
State the Theorem of the Continuous Inverse
Which proof is its proof analogous to?
Satz 3.3.2, pg 37
anal to satz 3.3.1 closed mapping thm
Define linear graph.
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.
Define extension and closable operator.
Def 3.4.2, pg 38
Prove A is closable iff ((xk,yk)) s.s.o. Gamma_A:
If xk–>0 and yk=Axk, then y=0.
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
Prove that a linear and contin operator is closable
(xk,yk=A=xk) in Graph_A s.t. xk–>0. Since A contin, |Axk|=0
Is A=d/dt closable?
Prove for 5+
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].
State the Lemma for the Calculus of Variations (“Fundamentallemma der Variationsrechnung”).
For 4: prove it
Satz 3.4.3, pg. 40
Define sublinear.
X R-VS. p:X–>R sublinear if 1) p(ax)=ap(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
State Hellinger-Töplitz
Prove for 4+
Beispiel 3.3.2, pg. 36
State the Hahn-Banach thm.
Sketch its proof for 3+
Satz 4.1.1, pg 43
Define C-sublinear
Additional property that R-sublin doesnt have? (that follows)
def. 4.1.2, pg 45, p>=0
Reformulate Hahn-Banach for C-vector spaces
Satz 4.1.2, pg 45
State and prove the theorem of “dominated extension” (“Dominierte Fortsetzung”)
Satz 4.1.3, pg 46
Define dualspace
(X,||.||) normed VS. X*:=L(X,R) dualspace of X.
For every x in X there is an x* in X*, s.t. ???
Prove for 3+
s.t. x(x)==||x||^2=||x||^2 (satz 4.2.1, pg 46)
For all x in X: sup{ |(x,x)| : ||x|| =< 1}=?
For all x* in X: sup{ |(x,x)| : ||x|| =< 1}=?
=||x||
=||x*||
i.e. takes the supremum
Satz 4.2.2, pg 47
Let x != y in X then ?
Prove for 2+
there is an l in X* s.t. l(x) != l(y)
Satz 4.2.3, pg 47
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+
Satz 4.2.4, pg 47
Define annihilator
def 4.2.2, page 48
Define j_y and J
What properties do they have?
pg 49, satz 4.3.1
State Riesz representation theorem (Rieszscher Darstellungssatz)
State strategy of proof for 5+
satz 4.3.2, pg 49
State the Lax-Milgram theorem,
its corollary for 5+
satz 4.3.3, pg 51
What can we say about the dualspace of L^p under certain (which?) conditions of the measure space?
satz 4.4.1, lemma 4.4.1 and lemma 4.4.2 pg 53
State the separation theorem for convex sets
satz 4.5.1, pg 56
Define the Minkowski-Funktional
pg 57
define extremal subsets and points
give two contrasting examples in R^2
def 4.5.1, pg 58
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+
Lemma 4.5.1, pg 59
Lemma 4.5.2, pg 59
State the Krein-Milman theorem (only pf for ii) for 5+)
satz 4.5.2 pg 59 and satz 4.5.3 pg 60
Define convex hull
def 4.5.2, pg 60
Define weakly convergent
definition 4.6.1, pg 60
If (xk)k converges weakly, what can we say about (xk)k?
Prove for 5+
Hint: liminf |xk|?
satz 4.6.1, pg 61
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?
definition 4.6.2, pg 61
lemma 4.6.1, pg 62
Define weakly sequentially closed
definition 4.6.3, pg 62
How do weakly closed, weakly sequentially closed and closed relate?
Prove for 4+
weakly closed => w.s.c => closed
Lemma 4.6.2, pg 62
Give a counter example that disproves that closed does not imply weakly sequentially closed
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)
What can we say about a convex set and the weak topology?
Pf for 4+
the weak closure is equal to the closure (satz 4.6.2, pg 62)
State Mazur’s lemma
satz 4.6.3, pg 63
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?
when dimX
Define bidualspace
X**=(X*)*=L(X*,R) is the bidualspace of X. def 5.1.1., pg 65
Define the canonical embedding of the dual space of a space into the space
state the properties of this embedding
def 5.1.1., pg 65
Define reflexivity of a normed space State some examples and non examples 1 ex for 3, 3 ex for 4, 4 ex for 5
definition 5.1.2, pg 65
Another non example L^inf (beispiel 5.1.2, pg 67)
If X is reflexiv then X is also…
Proof for 3+
complete (bemerkung 5.1.1, pg 66), reflexive (Satz 5.1.2.)
Relate X and X* by reflexivity.
Satz 5.1.2, pg 66
i) X*
ii) complete, reflexiv
Under what sufficient conditions is a subspace of a normed space reflexiv?
X is reflexiv, Y closed subspace of Y, then Y is also reflexiv.
Satz 5.1.3, pg 67
Define separability (+characterization)
Note M is metric space: definition 5.2.1/bemerkung 5.2.1, pg 68
Under which conditions are subsets of _____ spaces are separable?
Prove
Subsets of separable metric spaces are separable w.r.t. the induced metric
Satz 5.2.1. pg 68
Relate dual spaces and separability
Prove 1 for 3, both for 4
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
Define weakly convergent*
definition 5.3.1, pg 69
State the Banach-Alaoglu theorem
Proof for 4+
satz 5.3.1, pg 70
State the Eberlein-ˇSmulyan theorem
Prove for 3+
satz 5.3.2, pg 72
State the approximation theorem
Prove for 3+
Satz 5.3.3, pg 72
Define w.s.l.s.c. (weakly sequentially lower semi-continuous) “schwach folgen-unterhalb-stetig”
definition 5.4.1, pg 73
Define coercive
def 5.4.2, pg 74
State the variation principle (“Variationsprinzip”)
Prove for 3+
satz 5.4.1, page 74
Define dual operator
definition 6.1.1., pg 77
Relate an operator and its dual with an equation
Prove for 5+
||A||=||A*||
Satz 6.1.1, page 77
What holds for densely defined dual operator’s dual and “any” “extensions”?
Pf for 5+
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
State Banach’s closed range theorem
Satz 6.2.1, pg 79
Wonder if I should add the next Satz and lemma?
Define compact operator
Definition 6.2.1, pg 80
What holds for cpt operators and weakly convergent sequences?
Pf for 5?
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
State Arzéla-Ascoli
satz 6.3.1, pg 81
State the Fréchet-Kolmogorov theorem (if manageable)
satz 6.3.2, pg 82
Define adjoint operator
definition 6.4.1, pg 84
How to the dual operator and adjoint operator relate?
Bemerkung 6.4.1, pg 85
Define symmetric and self adjoint
definition 6.4.2, pg 85
Define resolvent set and spectrum
definition 6.5.1, pg 86
Define resolvent and describe how the resolvent and resolvent set relate. And give the topological properties of resolvent set and spectrum that it implies.
definition 6.5.2, pg 87
satz 6.5.1
Do two resolvents commute (w.r.t. the same operator)?
Yes (satz 6.5.2, pg 87)
Define eigenvalue, eigenspace, point spectrum, continuous spectrum and restspectrum
definition 6.5.3, pg 88
How do the spectral radius, the spectrum and the resolvent set relate?
Satz 6.5.3, pg 89
Define umrundet, i.e. surrounds and “schar”
definition 6.5.4, pg 90
State the spectral (mapping) theorem,
Pf for 4+,
(Lemma 6.5.2 may be used)
Satz 6.5.4, pg 91
What hold for the point spectrum when A is symmetric?
Pf for 4+
The eigenvalues are real, i.e. the point spectrum is a subset of R (satz 6.6.1, pg 92)
Define weak derivative
definition 6.6.1, pg 92
Give characterizations of selfadjointness
Satz 6.6.2, pg 95
Define normal and unitary
definition 6.7.1, pg 96
What follows for the spectral radius of a normal operator on a Hilbert space?
satz 6.7.1, pg 98
State Baire’s theorem on functions (long proof)
satz 1.4.1, pg 10
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
lemma 4.5.2, pg 59
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.
korollar 4.3.1, page 52
Define and characterize topologically complemented subspaces V s.s.o. X.
Prove why they are closed.
Exercise 3, sheet 4
