# Content of the lecture

### May 4th

Introduction; practical information; requirements for the grade. Please sign up for the Exercise class!

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)).

Chapter 1: Topological and metric spaces.

Start on recalling Euclidean topology on R^n (distance, open sets).

### May 6th

Continuation of (recalling) Euclidean topology on R^n: finite intersection of open sets and any union of open sets, are open; continuity of maps between R^n and R^m, by epsilon-delta, by open balls, and by open sets.

Definition of topology and topological space (the members of the topology are the open sets), discussion of discrete and in-discrete topology: Any set (with more than two points) can be equipped with more than one topology. Weaker/stronger topologies (not all topologies can be compared! - but discrete/indiscrete topologies can be compared to all topologies).

Definition of continuity of maps between topological spaces by open sets; definition of continuity AT a point. The identity map is continuous (using the same topology on the two copies!), any constant map is continuous, composition of continuous maps is continuous.

Definition of subspace/induced/relative topology on H (H subset A, {A,T} topological space).

(The material can be found in many books; for instance W A Sutherland, Introduction to metric and topological spaces. See also these lecture notes, where the order of things is different, however. There are some very good explanations in these notes by Prof. Erdoes. We will get to some of the concepts mentioned in these notes later.)

### May 11th

Definition of homeomorphism, and homeomorphic spaces. Definition of closed subset (examples given); the empty set, and the whole set are closed, the union of any number of finite closed sets is closed, the intersection of any number of closed sets is closed. Definition of neighbourhood. Definition of limit points (examples), and of closure (examples). A point x is in the closure of H iff every open set containing x has non-empty intersection with H.

H is closed iff it equals its closure. H subset K implies closure H subset closure K. closure of closure H equal closure H; closure H is a closed set.

Definition of (everywhere) dense subset of a topological space, and of separable topological space.

Definition of interior (examples), and of nowhere dense. H subset A is nowhere dense in T={A,T} iff A\closure(H) is dense in T. Corollary: A closed subset is nowhere dense iff its complement is dense. Definition of boundary of a set.

Definition of sequence, and of convergence of a sequence in a topological space. Claim that this definition can lead to non-unique limits of convergent sequences.

### May 13th

Example (indiscrete topology) of sequences with several limits. Definition of Hausdorff space; proof that in Hausdorff spaces limits of convergent sequences are unique. Any subspace of a Hausdorff space is Hausdorff. If f:A -> B is injective, and if B is Hausdorff, then A is Hausdorff.

Definition of metric spaces. Euclidean metric, discrete metric, metrics d_p on R^2, metric on C. Definition of B_r(x;d). Any metric space is a topological space. A topological space is metrizable if its topological arises from a metric. (This is not always the case (example; more to come)). Any notion for topological spaces (closed set, limit points, closure etc etc) therefore makes sense for metric spaces.

Every metric space is Hausdorff. Re-writing (using metric / B_r(x;d) ) definition of a dense subset. Definition of a bounded subset in a metric space.

### May 18th

Definition of diameter of bounded set; the union of finitely many bounded sets is bounded. Re-formulation of continuity (by open sets) of maps between metric spaces via the two metrics ('epsilon-delta definition'). Definition of bounded maps.

Definition of metrics d_p on R^n; Hoelder's and Minkowski's inequalities (proofs in tutorium). Definition of the sequence space l_p, l_infinity, and the metrics d_p, d_infinity on them.

Compactness: Definition of cover, open cover, subcover, compactness. Re-formulation for induced topologies on subsets of topological spaces (i.e. 'compact subset'). Heine-Borel: In R^n, compactness of a subset is equivalent to 'closed and bounded'. In general, this is not the case!

### May 20th

A compact subset of a metric space is bounded; a compact subset of a Hausdorff space is closed; a closed subset of a compact space is compact.

The image of a compact space by a continuous function is compact. A continuous map from a compact space to a metric space is bounded. A continuous map from a compact space into the reals (with the Euclidean topology) attains its bounds (max and min).

Definition of Cauchy-sequences in a metric space. Any convergent sequence is Cauchy (but not all Cauchy sequences are convergent). Definition of complete metric space. R^n with the Euclidean metric is complete (known).

A subset K of a metric space A is closed iff ( {a_n} in K, a_n -> a, a in A implies a in K ). If a Cauchy sequence has a convergent subsequence, then it is convergent (to the same limit).

Definition of sequentially compact (in itself/in M) of a subset C of a metric space M. A subspace C of a metric space M is compact iff it is sequentially compact (proof later!). Consequence: Any bounded (in Euclidean metric) sequence in R^n has a convergent (in Euclidean metric) subsequence.

Any compact metric space is complete. A complete subspace of a metric space is closed. A closed subset of a complete metric space is a complete metric space.

### May 25th

For X a non-empty set, M={A,d} metric space, let B(X,A) be the set of bounded maps from X to A. Then { B(X,A), d_{infinity} } is a complete metric space iff M is complete.

For M_1={A_1,d_1}, M_2={A_2,d_2} two metric spaces, let C(A_1,A_2) be the set of (d_1,d_2)-continuous maps from A_1 to A_2, and let C_b(A_1,A_2) be the set of maps from A_1 to A_2 which are (d_1,d_2)-continuous and bounded. C_b(A_1,A_2) subset C(A_1,A_2), C_b(A_1,A_2) subset B(A_1,A_2). If M_1 is compact, then C_b(A_1,A_2)=C(A_1,A_2) for all A_2.

{ C_b(A_1,A_2), d_{infinity} } is complete iff M_2 is complete. I.e., the uniform limit of a sequence of continuous and bounded maps is continuous and bounded.

Definition of uniform continuity. A continuous map from a compact metric space to a metric space is uniform continuous. Ex. f:[0,1] -> R.

Definition of totally bounded. For a subset C of a metric space the following is equivalent: (a) C is compact (b) C is sequentially compact (c) C is totally bounded and complete. (a) implies (b) proved.

### May 27th

Finished proof from last time ((b) implies (c), (c) implies (a); see above).

Arzela-Ascoli's Theorem:
Let A_1 be compact metric space, A_2 complete metric space. A subset M of C(A_1,A_2) is compact iff
(a) For all x in A_1 the set { f(x) | f in M } is compact in A_2
(b) M is equicontinuous
(c) M is closed.

### June 1st

Baire's theorem: The intersection of a countable family of open dense subsets of a complete metric space is dense.

Chapter 2: Banach and Hilbert spaces.

Motivation and discussion on Linear Algebra - the study of linear equations, matrices, vector spaces, linear maps. The axioms of vector spaces assumed known (Shall almost only study real or complex vector spaces).

Definition of semi-norms, norms, and the metric induced by a norm. Addtion of vectors, and multiplication by scalars are continuous operations with respect to this metric: The linear and the topologival structures are compatible.

Definition of Banach space.
Examples: R^n -- l_{infinity} -- l_{infinity}(M,Y) for M any non-empty set, Y Banach space -- C_b(A,X) for A metric space, X Banach space -- ( C^{alpha}, ||.||_{alpha} ) -- ( C^1, ||.||_{C^1} )

### June 3rd

Another example: l_p (proof given)

Definition of convex set, convex hull, absolutely convex set, absolutely convex hull, strictly normed/strictly convex space, uniformly convex space, open unit ball, unit sphere. Closure of open unit ball is "closed unit ball"; both are convex.

Definition of linear subspace and of quotient space. Discussion about linear/topological subspaces. Given a semi-normed space, the null-set of the semi-norm is a linear subspace, and dividing out and using the semi-norm on representatives gives a norm on the quotient space.

### June 8th

The 1- and infinity norms on C[0,1] are not equivalent. Discussion.

Definition of (equivalent) l_p norms on the direct sum of two normed spaces. Gives a Banach space if each component is Banach.

Definition of distance between a point x and a subset U of a metric space A, and of a "best approximation" a to x in A. This may not exist, and if it does, may not be unique. If U is compact, there is at least one.

Riesz' lemma: For a non-trivial closed subspace U, and d in (0,1), there exists x_d of length 1 s.t. ||x_d - u || >= 1-d for all u in U.

Definition of linear maps (called linear operators), kernel, and image of linear maps. A linear operator is continuous at a single point iff it is bounded on the unit ball iff it is uniformly continuous iff there is a C such that ||Tx|| =< C ||x|| for all x. In this case, T is called a bounded linar operator (Discussion on connection to bounded maps). Not all linear maps are bounded: Example of a discontinuous linear map given (d/dx from C^1 to C, both with sup-norm).

### June 10th

More remarks: If T:X -> Y bounded, and one choses equivalent norm(s), then T remains bounded (but the operator norm of T might change). Any linear map from a finite dimensional space is bounded.

The set B(X,Y) of bounded linear maps is a vectorspace, and the operator norm IS a norm. Furthermore, ||ST|| <= ||S|| ||T||. If X is K vectorspace, then X'=B(X,K) is called the dual space (space of bounded linear functionals). Examples of various linear bounded functionals.

If Y is Banach, then B(X,Y) is Banach; in particular, X' is always Banach.

Definition of isomorphism, isometry on/in Y, and of isomorphic/isometrically isomorphic normed space. These are both equivalence relations. Any two normed spaces of same finite dimension are isomorphic, but not necessarily isometrically. Definition of embedding, and (linear) projections.

### June 15th

More remarks/definitions: Invertible operator. For bounded maps between infinite dimensional space, surjectivity OR injectivity alone not enough to get bijectivity.

Unique extension of a bounded operator from a subspace V (of X) to a Banach space Y, to a bounded linear operator on the closure (in X) of V (still to Y), and with same norm. Example: When V is dense in X. If original operator an isometry, then also extension. However, injectivity may not be preserved. Extension of bounded linear functionals.

Definition of scalar (inner) product, pre-Hilbert space. Scalar product gives a norm; Cauchy-Schwarz' inequality, parallelogram rule. Definition of a Hilbert space. Polarization identity. Criterion for a norm to come from a scalar product: It has to satisfy (it is enough it satisfies) the paralellogram rule. Continuity of the scalar product. Examples (to be continued).

### June 18th

Examples of (pre)-Hilbert spaces: C^n (H), l_2 (H), C[0,1] (with integral of product f, g) (pre-H), C^k (sum of same for all derivatives) (pre-H).

Definition of orthogonality of vectors (and Pythagoras), and orthogonality of subsets. Definition of orthogonal complement A^perp of subset A. This is always a closed subspace, and the orthogonal complement of the closure of A is A^perp. A subset (A^perp)^perp.

For a closed and convex subset of a Hilbert space H, and any x_0 in H, there is a unique best approximation (called P(x_0)) to x_0 in K. P is a projection (P^2=P). P(x_0) can be characterised by
Re < x_0-P(x_0),y-P(x_0) > <= 0 for all y in K.

Theorem on the orthogonal projection on a closed linear subspace U of a Hilbert space, and its properties; H can be written as an (orthogonal) direct sum of U and U^perp.

### June 22nd

The closure of a linear subspace U of a Hilbert space equals (U^perp)^perp.

Frechet-Riesz' Representation Theorem: The dual of a Hilbert space H is isometrically (conjugate) isomorph to H.

Definition of algebraic basis of a vector space. Every vector space has a basis (using Zorn's Lemma: Every totally ordered subset of a partially ordered set has a maximal element). However, the number of elements on a basis is either finite or uncountable. Hence, useless (!) for infinite dimensional vector spaces.

Definition of a 'family' (a map from an index set I to a Banach space X - generalisation of sequence). Definition of absolutely summable families, and of l_1(I). Definition of support of a family, and of the sum of a family in l_1(I). Definition of square summable families and l_2(I); this is a Hilbert space. Definition of orthonormal system (ONS), and of a maximal ONS. Examples. Definition of Fourier-coefficients wrt. an ONS.

### June 24th

Bessel's inequality. Expansion of elements wrt. ONS and convergence. The Fourier-map wrt. an ONS in a Hilbert space. Equivalence of (a) The expansion of x wrt to the ONS converges to x (b) Parseval's identity (c) The Fourier-map wrt the ONS is an isometry (d) The linear span of the ONS is dense in H (e) The Fourier-map is injective (f) The ONS is maximal.

An ONS satisfying the above is called complete, or an orthonormal basis (ONB) for H (a Hilbert space basis). If dim H is infinite it is NOT an algebraic basis! Every Hilbert space has an ONB (using Zorn's Lemma). Every ONB of H has the same cardinality. If the ONB is countable, then H is separable. In this case, H is isometrically isomorph to l_2(N).

### June 29th

Chapter 4: Measures, integration and Lp-spaces.

Discussion of the 'measure-problem' (existence of sigma additive, translation and rotational invariant measure on ALL subsets of R^n) (Vitali, Banach-Tarski). Definition of sigma-algebra and examples.

Definition of measure, and examples (counting measure, Dirac measure). Construction of measures: Definition of semi-ring; example: 'boxes' in R^n. Definition of content and pre-measure; examples (Lebesgue content in R^n; Lebesgue-Stieltje content, for a monotone increasing function F).

How to get from pre-measure via exterior measure to a measure. Definition of measurable sets. Caratheodory's extension theorem. Example (Lebesgue measure on R^n) and discussion. Definition of sigma-finiteness of a content or pre-measure. Uniqueness of extension of a sigma-finite pre-measure to a measure. Example: Lebesgue pre-measure on R^n and Lebesgue-Stieltje pre-measures are sigma-finite.

### July 1st

Definition of measurable functions; (almost) every combination of measurable functions is measurable (sum, product, max/min, sup/inf, lim inf/lim sup, hence, pointwise limit). A positive valued function is measurable iff it is the pointwise limit from below of step functions (functions with only finitely many values). Continuous functions are measurable wrt. Borel sets.

Definition of integral (and of "f is integrable") for positive measurable step functions, positive measurable functions, real valued functions, and complex valued functions. Definition of "almost everywhere" (a.e. wrt measure mu). If the integral of f (non-negative and measurable) is 0, then f is 0 mu-a.e. If f=g mu-a.e. for integrable f, g, then their integrals are equal. If f is Riemann integrable on [0,1], then it is Lebesque-integrable (integrable wrt lambda^1 on [0,1]).

Definition of space of p-integrable functions, and the p-(semi)norm. Definition of the space L^p by defining f to be equivalent to g iff f=g mu-a.e. Discussions. Proof that L^p is a vector space, and that the (semi)norm IS a (semi)norm (to be continued).

### July 6th

Hoelder's and Minkowski's inequalities in L^p, completeness of L^p (using Beppo-Levi's Thm/Lebesgue's Thm on Monotone convergence, and Lebesgue's Thm on Dominated Convergence). Note: If X=natural numbers, sigma-algebra equal power set, and measure equal counting measure, we get l_p.

Defintion of L^infty, essential supremum, completeness, Hoelder's inequality.

Discussion on bounded linear functionals.

### July 8th

Definition of sublinear functionals on a real vector space.
Hahn-Banach's theorem: A linear functional f on a subspace V_0 of a real linear space E, which is bounded by a sublinear functional p can be extended to all of E, maintaining the bound.
Hahn-Banach for functionals bounded by semi-norms, on a K-vector space.
Hahn-Banach for linear functionals: extension to the whole vector space.
Corollaries: For non-zero elements x, there is an f in X' such that f(x) not eq 0. In particular, X' separates points in X.

### July 13th

Banach-Steinhaus/Principle of Uniform Boundedness.

Defintion of open map. Remarks. Lemma: T open iff there is a ball contained in T(B_1(0)). Open Mapping Theorem (pf to be continued). Corollories: T in B(X,Y) bijective, then T^{-1} bounded. Equivalence of norms. T in B(X,Y) injective, then T^{-1} (from R(T)) bounded iff R(T) closed.

### July 15th

Open Mapping Theorem (end of proof).

Definition of closed operator. Discussions on relationship to continuity. Definition of graph gr(T), and of graph norm. Lemma: (1) graph linear subspace, (2) T closed iff gr(T) closed in X x Y (with l_1-norm). Lemma: T closed, then (1) domain with graph norm is Banach (2) T is bounded from D with graph norm.

Theorem: T closed and surjectiv implies T open, if also injective, then T^{-1} bounded.

Closed Graph Theorem. Discussions. Theorem: The pointwise limit of bounded operators is a bounded operator.

Computing the norm of x in X as max over |f(x)| for f in X', ||f|| le 1. The bi-dual X'', and the canonical embedding of X in X''.

### July 20th

Definition of weakly bounded subsets. Weakly bounded implies norm bounded. Definition of reflexiv space. Any Hilbert space is reflexiv. Any reflexiv space is complete. Reflexivity is preserved among isometrically isomorphic spaces. X, Y reflexiv implies X x Y reflexiv. Ex: l_p reflexiv for 1 < p < infinity (seen). Ditto for L^p (no proof yet).

Definition of weak convergence (in X), and of weak* convergence (in X'). Similarly for weak and weak* Cauchy. Definition of weakly sequentially compact, and same for weak*. These notions can be defined via a topology, but not given here (yet). Convergence in the norm is called strong convergence.

Remark: Weak limits are unique (using Hahn-Banach). Strong convergence implies weak and weak* convergence. Opposite not true (ex: {e_n} in l_p). The norm (in X) is lower semi-continuous wrt. weak convergence, and ( the norm in X') wrt. weak* convergence. Weak and weak* convergent sequences are bounded.

For X separable, the closed unit ball in X' is weak* sequentially compact. (proof next time).

### July 22nd

Proof that, for X separable, the closed unit ball in X' is weak* sequentially compact.

Definition of the topology behind weak convergence, and discussions. Similarly for weak*.

On reflexivity: For a reflexiv space, weak (in X') and weak* (also in X') convergence is the same. If V subset X, closed linear subspace, X reflexiv, then V reflexiv. X reflexiv implies X' reflexiv. X' separable implies X separable (no proofs).

Banach-Alaoglu: In a reflexiv Banach space, the closed unit ball is weakly sequentially compact. Special case: Any Hilbert space. Discussions.

Chapter 5: Topics in bounded operators.

Definition of compact operators. Remarks and discussions. Example: Integral operators (defined through continuous kernels) on C[0,1] (Problem 24).

### July 25th

Definition and properties of (Banach space) adjoint operator, definition and properties of (Hilbert space) adjoint operator.

Definition of resolvent set, resolvent (operator), and spectrum of an operator. Splitting of spectrum in point, continuous, and rest spectrum. Discussion.

The resolvent set is an open subset of C, and the resolvent function is an analytic map. Nuemann-series. The set of invertible operators is open in B(X,Y).

Functions of operators: Polynomials, analytic functions (power series). Examples. Discussion of FA2 and spectral theory.

### July 27th

(This lecture an overview/discussion only - no proofs given.)

Definition of Fredholm operator and index. I-K is Fredholm if K is compact operator. Spectral Theorem for compact operators (on Banach space!) - statement and discussion (proof in FA2). Consequence for resolvent function: isolated poles of finite order. Fredholm Alternative and discussion.

Schauder: X is compact iff X' (adjoint) is compact.

Spectral Theorem for normal operators on Hilbert space.

Last update: October 11th., 2012 by Thomas Østergaard Sørensen.