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.