### Vorlesung: Funktionalanalysis (FA1) (SoSe 2016)

Content of the lecture (Kurzübersicht der Vorlesung):

In-official, non-corrected, and not necessary up-to-date TeX'ed version of Martin Peev's notes from the lecture can be found here. (For typos etc, email martin.peev-a-t-hotmail.com.)

12 April:

Introduction; practical information; requirements for the grade (see main page).

Motivation: Functional Analysis (FA) is 'infinite dimensional linear algebra' - the study of infinte dimensional vector spaces and of linear maps between them. It is a 'fusion' of Linear Algebra (LA) and Analysis.

(From LA, the following are assumed to be known: vector, matrix, linear (in)dependence, span, basis, linear map and its matrix, range, kernel, rank, Gauss elimination, scalar product, positive definite matrix, norm of a vector and a matrix, orthonormal basis, orthogonal and unitary matrix, orthogonal projection, change of basis formula.)

Both of these - LA and Analysis - are very much about studying equations and their solutions (example given: linear systems, eigenvalue eq, Intermediate Value Thm).

Lots of the motivation in FA comes from studying equations - mostly differential and integral eq's. Example of general ODE (Gewoehnliche Diff Gleichung) and of 'integral operators' (linear map K given by integrating (in t) function x(s) against a function k(t,s) of two variables (s,t)).

14 April:

End of: Motivation (from last time).

'Speech' on: Writing Mathematics properly, and how we plan to teach you!

Chapter 1: Topological and metric spaces.

Recall on Euclidean topology on R^n (distance, open sets, continuity, by epsilon-delta, and by open sets).

Definition of topological space, Hausdorff space. Definition of stronger and weaker topologies. Remarks (discrete topology).

19 April:

Remarks: The discrete is stronger, the indiscrete weaker than any other topology.

Definition of relative/induced topology.

Definition of closed sets, the interior of a set, the closure of a set. The interior is open, the closure is closed. Discussion of what happens in discrete/indiscrete case.

Definition of neighbourhood of a point, inner point of a set, adherent point of a set, limit point of a set, boundary points of a set, boundary of a set.

Lemma: A set is open iff all its points are inner points. A set is closed iff it contains all its adherent points. The interior of a set is the set of all its inner points, and the closure of a set is the union of the set and its limit points.

Definition of convergence of a sequence.

Definition of continuity of a map between two topological spaces. Definition of open map.

Discussion of convergence of a sequences. Definition of homeomorphism, and homeomorphic top. spaces.

Definition of a neighbourhood basis of a point, and of first countable top. space.

April 21st

Definition of sequentially continuous.

A continuous map f:X -> Y is seq. cont. IF X is first countable, and f is seq. cont., then f is cont.

Definition of local basis for x (a point). Definition of how to use a local basis (one for _each_ x in X) to generate a topology.

Definition of a topology generated by _any_ family of subsets. Definition of subbasis and basis. Definition of second countable. The Euclidean topology on R is second countable - the family of open intervals with rational endpoints is a countable basis.

Definition of a dense subset, and of a separable top. space.

April 26th

Forgot last time:
Let X be first countable, let A be subset of X, x adherent point of A. Then there is a sequence of points _in_ A, which converges to x. In particular, the closure of A equals the set of points which are limits of a sequence of points from A.

Correction of mistake from last time: Formulation of Theorem 1.19.

Definition of the product topology on X x Y. Comments, in particular: In the product topology, convergence is "coordinate wise".

Definition of metric d, and metric space (X,d). Some further inequalities (e.g. inverse triangle). Definition of the distance from a point to a subset, and between two subsets of X. Examples: Euclidean ('2-metric') on R^n, and the 'p-metrics' on R^n, the discrete metric, the induced metric on a subset, the French railroad metric. Definition of B_R(x), and K_R(x), of bounded sets, and of the topology T_d generated by {B_{1/k}(x)} for k in N.

Remark: The sets B_{1/k}(x) (and B_R(x)) are open. Any metric space is first countable - hence, a map (out of a metric space, into a topological space) is continuous iff it is sequentially continuous. Translation of convergence of sequences using the metric. Note, the closure of B_R(x) _might_ not be K_R(x) ! The metric (as a function of two variables) is continuous.

Definition of Cauchy sequence, and of a complete metric space. Remark: Every convergent sequence is Cauchy.

April 28th

Remark: Every Cauchy sequence is bounded.

Lemma: (X,d) complete. Subset A of X is a complete metric space iff it is a closed subset of X.

Definition of isometry and of isometric metric spaces. Remarks.
Theorem: Completion of a metric space (unique up to isometry).

Discussion of fixpoint equations, and fixpoints. Definition of Lipschitz continuous, Lipschitz constant, contraction. Definition of uniformly continuous. Remarks: Lipschitz implies uniformly implies continuous. Hence, contraction is continuous.

Formulation and discussion of Banach's Fixpoint Theorem. Proof of Banach's Fixpoint Theorem. (End of proof next time.)

May 3rd

End of proof of Banach's Fixpoint Theorem.

Definition of compact, sequentially compact, relative compact subsets of a topological space. Discussion: In R^n compact equivalent to closed & bounded, in general NOT.

Lemma: A continuous real map on a sequentially compact space attains its minimum (and maximum).

Theorem: If a space is first countable and compact, then it is sequentially compact.

Proposition: A compact subset of a Hausdorff space is closed. Proposition: A closed subset of a compact space is compact. Corollary: In a compact Hausdorff space, subsets are compact iff they are closed. (Proofs next time).

May 10th

(From last time) Proof of: A compact subset of a Hausdorff space is closed. A closed subset of a compact space is compact.

Proposition: A sequentially compact metric space is complete. (Lemma: A Cauchy sequence with a convergent subsequence, is itself convergent, with same limit).

Definition of pre-compact.

Theorem: Let (X,d) be complete metric space. A subset A is compact iff it is sequentially compact. A _closed_ subset A is pre-compact iff it is compact.

May 12th

Theorem: Any compact metric space is separable. Theorem: X, Y topological spaces, f: X-> Y map. If X is compact and if f is continuous, then f(X) is compact (in Y). Corollary: A continuous real-valued function on a compact topological space attains its maximum and minimum. Discussion.

Theorem: (X,T) compact Hausdorff space. If T_w is any topology strictly weaker than T, then (X,T_w) is not Hausdorff. If T_s is any topology strictly stronger than T, then (X,T_s) is not compact.

For more on general topology, see for example
W. A. Sutherland: Introduction to metric and topological spaces, Oxford University Press.
J. L. Kelley: General Topology, Springer Verlag.
G. K. Pedersen: Analysis Now, Springer Verlag.

Chapter 2: Banach and Hilbert spaces.

Definition of norm, normed space, semi-norm. Remarks: Definition of metric given by norm. The norm is continuous (from X to R) (with respect to topology given by metric). Any convergent sequence is bounded. The linear structure (vector addition and scalar multiplication) is continuous. The balls {B_{1/k}(0)}_{k in N} form a countable neighbourhood basis of 0 (the zero-vector in X). One gets a countable neighbourhood basis of any x in X by translation: B_R(x) = x + B_R(0).

May 19th Definition: A complete normed space is called a Banach space.

Examples: R^n with usual Euclidean norm. R^n with p-norm.

Example: l_p, with p-norm (p in [1,infinity]), is a normed space. It is a Banach space.

Definition: Equivalent norms on a normed space. Example: Different l_p - norms on R^n.

Theorem: Any two norms on a finite dimensional normed vector space are equivalent. Any finite dimensional normed vector space is a Banach space. Any finite dimensional linear subspace of a normed vector space is a closed subset, and is complete (even if larger space is not).
(End of proof next time.)

May 24th

End of proof from last time (see above).

Riesz' Lemma: X normed space, U proper closed linear subspace, lampda in (0,1). There exists vector x of norm 1, such that ||x-u|| >= lambda for all u in U.

Theorem: The closed unit ball in a normed space is compact iff the space is finite dimensional.

Definition of linear maps between normed spaces, and of their kernel and range (which are linear subspaces). The kernel is closed if the map is continuous.

Theorem: A linear map is continuous iff it is continuous at 0 iff it is a bounded linear operator (there exists C>0 sucht that ||Tx|| <= C ||x|| ). (End of proof next time.)

May 31st

End of proof from last time (see above)

Definition of operator "norm", and of B(X,Y). Lemma on facts on the operator "norm".

Any linear map out of a finite dimensional normed space into a normed linear space is bounded/continuous.

Theorem: B(X,Y) with the operator "norm" is a normed space (ie, the operator "norm" is a norm). If Y is Banach, then so is B(X,Y) with the operator norm.

Definition of the dual space of a normed vector space and its norm. The dual space is always a Banach space (with the operator norm).

Examples: The dual of any finite dimensional normed space is (isometrically isomorphic to) itself.

The dual of l_p (1 < p < infinity) is (isometrically isomorphic to) l_q (q the conjugate exponent) (end of proof next time).

June 2nd

End of proof from last time.

Theorem: X, Y normed spaces, Y Banach, M dense linear subspace of X, T:M -> Y linear and bounded. Then T has a unique extension to a linear bounded map T-tilde:X -> Y, with the same operator norm.

Definition of a compact map between normed spaces, and of a compact operator, and of K(X,Y). A compact operator is bounded, so K(X,Y) subset B(X,Y). Definition of continuous, compact, and dense embeddings. Discussion. The composition of a compact and a bounded operator is again compact (proof next time).

June 7th

Proof from last time (see above).

Definition of inner product/scalar product, inner product space. Remarks. Cauchy-Schwarz-Bunyakowski's inequality. The inner product generates a norm. The inner product is continuous in this norm. The Parallelogram Identity. Definition of a Hilbert space.

Examples: R^n,C^n (usual scalar products), R^n with _any_ scalar product, l_2, C[0,1] with < f,g >=int \overline{f(x)} g(x) dx (NOT Hilbert space).

The scalar product with a fixed vector y defines a bounded linear functional. Conversely: Riesz' representation theorem: Every bounded linear functional f on a Hilbert space is given in this way. The vector y also minimizes F(z)= < z,z > - 2 Re f(z). (End of proof next time).

June 9th

End of proof from last time (Riesz' representation theorem).

Definition of orthogonal vectors, and orthogonal complement of a set.

The Projection Theorem.

Definition of boundedness and coercivity of a sesqui linear form on a normed space.

Lax-Milgram's Theorem: For any bounded, coercive sesquilinear form b on a Hilbert space there is a bounded linear bijection R such that b(Rx,y) = < x, y > for all x,y in H. (NO PROOF!)

Discussion of algebraic basis for a vector space: Definition of algebraic (Hamel) basis. Theorem: Every vector space has a Hamel basis. Proof follows from Zorn's Lemma. Proposition: Let X be a Banach space which is NOT finite dimensional, and let B be any Hamel basis for X. Then B is uncountable. This makes the concept of a Hamel basis useless (for us!).

June 14th

For general index set I, and Banach space X: Defintion of family x:I -> X, and of absolutely summable families. Definition of support of an absolutely summable family (a countable subset of I). Definition of square absolutely summable families, and of the Hilbert space l_2(I).

Definition of an orthonormal systen (ONS) and a maximal ONS, in inner product space. Examples. Definition of Fourier-coefficients x^(i) of an element x in H with respect to an ONS.

Lemma: Pythagoras. Bessel's Inequality.

Given an ONS (indexed over I) for H, and an element x in l_2(I), definition of sum x_i e_i, as element in H.

Definition of the Fourier-map, taking x in H into the family of its Fourier coefficients (an element of l_2(I)). It is linear, bounded, surjective, with operator norm less or equal 1.

Theorem: An ONS is maximal iff the Fourier map is injective iff the Fourier map is an isometry iff the linear span of the ONS is dense in H iff Parseval's Identity holds iff for all x, the element sum x^(i) e_i equals x. (End of proof next time).

June 16th

End of proof of the equivalent characterizations of an ONB. Every Hilbert space has an ONB and is isometrically isomorphic to l_2(I), but this is constructive only if the space is separable.

Chapter 3: (Some) function spaces.

The space C(X) of continuous functions over a compact metric space X. Theorem: C(X) equipped with the supremum norm is a Banach space. The proof is a "3epsilon argument" (useful trick).

Theorem (Ascoli-Arzela'): a bounded and equicontinuous subset of C(X) is relatively compact. In the equivalent language of sequential compactness a uniformly bdd and equicont. family of continuous functions on X admits a subsequence that converges uniformly. (The proof (not given/not given yet) is a "diagonal trick" (another useful trick)).

Definition of Hoelder continuous functions with exponent gamma (Lipschitz when gamma=1) with their natural norm.

Theorem: Hoelder cont. functions form a Banach space. The spaces of gamma-Hoelder continuous functions (on a bounded domain) embed compactly into each other when gamma increases. (They are nested, the larger the gamma the smaller the space.) (Proof (not given): Uses Arzela-Ascoli).

Overview of L^p spaces. (See Prof. Mueller's Analysis III lecture as well as Prof. Griesemer's survey notes). Notions assumed to be well-known: sigma-algebras, measurable functions, def of integration. In the following we shall recall the L^p theory without proofs.

Definition of: p-seminorms for p between 1 and infinity, endpoint included. The case p=infinity involves the essential supremum.

Taking equivalence classes with respect to equality of functions almost everywhere, we turn the p-seminorm into a norm. We thus define the normed vector spaces L^p. Rigorously speaking they consists of equivalence classes. We will allow ourselves a standard abuse of notation.

(Riesz-Fischer:) L^p with p-norm is a Banach space. In particular, L^2 with the scalar product < f,g > = int \overline{f} g is a Hilbert space.

Theorem: Dual of L^p(X,mu) is isometrically isomorphic to L^q(X,mu), with q the conjugate exponent WHEN 1<= p < infinity AND (X,mu) is sigma-finite. That is, any bounded linear functional on L^p is simply integration against some L^q-function. (No proof (yet); uses Radon-Nikodym)).

June 21st

Chapter 4: The cornerstones of Functional Analysis

Definition of no-where dense and meagre sets, and of sets of first and second category.

Baire's category theorem: The intersection of countably many open and dense subsets of a complete metric space is itself dense.

Corollary: A complete metric space is of second category (in itself). Corollary: In a complete metric space, if the union of countably many closed sets contains an open ball, then at least one of the sets must contain an open ball too.

Principle of Uniform Boundedness / Banach - Steinhaus Theorem: For X Banach, Y normed, and a subset H of B(X,Y) which is _pointwise_ bounded, H is in fact uniformly bounded. Corollary: The pointwise limit of a sequence of bounded linear operators is a bounded linear operator.

Open Mapping Theorem / Inverse Mapping Theorem: X, Y Banach, T in B(X,Y) surjectiv. Then T is an open map. In particular, if T is a bijection, then T^{-1} is bounded (ie in B(Y,X)). (Re-cap and end of proof next time).

June 23rd

Re-cap and end of proof of Open Mapping Theorem. Corollary: If two norms on a space X both makes X a Banach space, and if one is bounded by the other, then they are equivalent.

Definition of graph G(T) of a map T, and of norm on X x Y for X, Y normed spaces. If X, Y are Banach, so is X x Y (seen earlier). Closed Graph Theorem: A linear map T:X -> Y (X, Y Banach) is bounded iff G(T) is closed (in X x Y). Discussion of the connection and difference between proving "T continuous" and "G(T) closed".

Definition of sublinear functionals.

Hahn-Banach Theorem (extension version): X a _real_ vector space, p a sublinear functional on X, M linear subspace of X, f linear functional on M, bounded pointwise (on M) by p. Then there exists an extension F (of f) to X, F a linear map, bounded by p (on X).(End of proof next time).

June 28th

End of proof last time (applying Zorn's Lemma).

Theorem: X K-vector space, M linear subspace, p semi-norm on X, f linear map on M, with |f| =< p on M. Then there exists extension F (of f) to X, F linear, with |F| <= p.

Corollary: X normed K-vector space, M linear subspace, f:M -> K linear and bounded. Then there exists F in X' which extends f, and the norms ||f||_{B(M,K)} and ||F||_{B(X,K)} are equal.

Theorem (Hahn-Banach separation theorem): X normed K-vector space, A,B disjoint convex subsets of X, A open. Then there exist F in X', and real number gamma so that Re F(x) < gamma <= Re F(y) for all x in A, y in B. Discussion: The set where F(z) = gamma (an affine subspace of co-dimension 1) separates A and B. (End of proof next time).

June 30th

End of proof of Hahn-Banach separation theorem.

Theorem: X Banach, M closed linear subspace, x not in M. Then there exists F in X' with norm 1, value at x equal dist(x,M)>0, and F restricted to M is zero. Consequence: For all non-zero vectors x in a Banach space, there is an element in X' with norm one such that F(x) = ||x||. Therefore, X' separates points in X.

Definition of bi-dual X'' of normed space X. The bi-dual is always Banach. Examples. Definition of canonical embedding: isometric embedding of X into X'' (via X', using H-B). Discussion of completion of non-Banach X via X'' and canonical embedding.

Definition: weakly bounded. Proposition: Weakly bounded implies norm-bounded (via Banach-Steinhaus).

Definition: Reflexive normed space. Examples and discussion. Definition of weak convergence (in normed space) and weak-* convergence (in a dual space), of weakly Cauchy, weak-* Cauchy, and of weakly sequentially compact and weak-* sequentially compact subsets. Discussion of difference between weak and weak-* convergence on the dual space X' (these two are the same if X is reflexive space).

July 5th

Recall: Definition of weak and weak-* convergence. Definition of the 'weak topology' (the topology giving rise to weak convergence). Discussion. 'Usual' (norm) convergence is called strong convergence.

Remarks: The weak limit is unique (via Hahn-Banach). So is the weak-* limit. Strong convergence imply weak convergence. The opposite is not true (take ONB in Hilbert space). If X is reflexive, V closed linear subspace, then also V is reflexive. X is reflexive iff X' is reflexive. If X' is separable, then X is separable (converse not true). The norm is lower (sequentially) semi-continuous with respect to weak convergence, as well as with respect to weak-* convergence. Discussion: Importance for minimization. Weak-convergent, and weak-* convergent sequences are norm-bounded.

Theorem: X separable. Then the closed unit ball in X' is weak-* sequentially compact (ie, any bounded sequence in X' has a weak-* convergent subsequence).

Theorem (Banach-Alaoglu): X reflexive Banach space. Then any (norm-) bounded sequence in X has a weakly convergent subsequence (ie. the closed unit ball in X is weakly sequentially compact).

July 7th

Proof of:
Theorem (Ascoli-Arzela'): a bounded and equicontinuous subset of C(X) is relatively compact. In the equivalent language of sequential compactness a uniformly bdd and equicont. family of continuous functions on X admits a subsequence that converges uniformly. (The proof is a "diagonal trick").

Definition: absolutely continuous measure. Radon-Nikodym derivative (or density) of an absolutely continuous measure.

(See Prof. Mueller's Analysis III lecture as well as Prof. Griesemer's survey notes).

Proof of:
Theorem: Dual of L^p(X,mu) is isometrically isomorphic to L^q(X,mu), with q the conjugate exponent WHEN 1<= p < infinity AND (X,mu) is sigma-finite. That is, any bounded linear functional on L^p is simply integration against some L^q-function. (Proof uses Radon-Nikodym; end of proof next time).

July 12th

End of proof from last time (see above). Remarks, discussion.

Chapter 5: Topics on bounded operators.

(This chapter contains less (!) proofs)

Recall: Definition of compact operators.

Favorite examples of compact operators: compact embeddings; certain integral operators.

Definition of (Banach space) adjoint of a bounded operator. Definition of Hilbert space adjoint of bounded operator. Discussions, and algebraic properties.

July 14th

The inverse of a bounded operator exists iff the inverse of the adjoint exists.

Definition of resolvent set, spectrum, point spectrum, continuous spectrum, rest spectrum for bounded operator on a Banach space (T in B(X)). Discussions and remarks. Definition of eigenvalue (point spectrum) and eigenvector, and discussion of relation (similarities/differences) to finite dimensional case.

Definition of resolvent map. The resolvent map is a complex analytic map on the resolvent set (with values in B(X)).

Thm: If X Banach, T in B(X) with norm < 1, then I-T is invertible, and it is given by a convergent power series (the Neumann series) a la for the geometric series. Consequence: The set of invertible operators is an open subset of B(X,Y).

Definition of f(T) for T in B(X) and f monomial, polynomial, convergent (complex, complex valued) power series. Examples and discussion.

Theorem (Schauder): T in B(X,Y) is compact iff T' in B(Y',X') is compact (no proof).

Eksempel: 'Diagonal' operator on a Hilbert space (sum of weighted projections). The operator is normal (TT^* = T^*T). If the weights go to zero, the operator is compact.

Spectral Theorem for compact, normal operators on a Hilbert space (no proof): Any such operator is of the form in the example.

End of lecture!

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Letzte Änderung: 19 July 2016 (no more updates).

Thomas Østergaard Sørensen

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