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
