TENTAMEN ONLINE: SUCCES! THIS is the general information page about the FUNCTIONAL ANALYSIS course in Blok 5, 2007. The course is intended for bachelor students in mathematics, physics and econometrics, but everybody else is welcome (within limits of course). CREDITS: 6 ECTS EXAMINATION: take home exam available on this site June 27 HOMEWORK SET 1: (see blackboards from Thursday) 1. X real normed space. Show that X is an inner product space if the the parallelogram law is satisfied. 2. H Hilbert space. K convex, closed nonempty subset. a in H. Show there exists a unique y in K such that d(a,K)=||y-a||. 3. Refering to 2 above. Show that the map P: H -> K defined by P(a)=y satisfies ||P(a)-P(b)|| is less or equal than ||a-b||. SIGN UP: LET ME KNOW BY E-MAIL. MATERIAL: I have decided to use the Karen Saxe book, Beginning Functional Analysis ISBN-0387952241. I just ordered a copy from Amazon that ships next week. There were lots of other copies available. Aydin just walked in, he will order 8 copies from Scheltema. Price 20 euro. Last year I used the first part of Robinson's book on Infinite-dimensional Systems of PDE's, the same book was used in the national mastermath PDE course. This book is no longer used for either of the two courses. I put some old material in this directory. fa.pdf is a handout I used in Leiden. fa2006.pdf is additional material from last year. e.pdf is material I used for a short course at the Lorentz Center on Spectral Analysis. COURSE HOURS: Tuesdays 10-12.45 in F131, Thursdays 13.30-16.15 in R223. First session April 3, no session on May 17, last session May 24. WHAT IS FUNCTIONAL ANALYSIS? Functional analysis is a toolkit for solving equations in which the unknowns are functions rather than numbers. Most of the equations we solved in analysis and linear algebra required finding a solution as a number or a finite set of numbers, which, substituted in some given function, would make it zero, or would maximise or minimise it. Consequently we learned in linear algebra and analysis all sorts of things about linear and nonlinear functions defined on subsets of R^m and C^m, finite dimensional vector spaces over the real or complex numbers, equipped with a natural (inner product) norm. Our lifes were made easy by the fact that bounded closed sets in R^m and C^m are compact so that bounded sequences have convergent subsequences. Another fact taking completely for granted was the continuity of linear functions. In the infinite-dimensional setting needed to solve problems such as the integral equations above, we first need good normed vector spaces in which our solutions are to be found. We will see many different possibilities to assign a norm to a function, leading to different spaces. There are many candidates for R^infty so to speak. This will make the theory of even only linear functionals a subtle issue in which linear algebra and analysis (epsilons and delta's) merge. PROGRAMME (is dynamic) Questions,answers. Answers, questions. Every opinion stated/suggested below is to be challenged. CHAPTER 1. The standard spaces reviewed. R^n standard n-dimensional real space C^n standard n-dimensional complex space both with standard inner product norm What are the basic properties of R^n used throughout the study of real valued functions on R^n? Answer involves the notions of compactness, completeness and dimension. What are open sets in R^n? (Why) do we need open sets in R^n? The standard norm is called a norm because of the 4 conditions (page 5,i-iv). Are there other norms on R^n and C^n? Yes. Ex 1.1.1,2. Are they really different? No? (First question to be done in detail). In what sense? See Ex 2.3.2-3. What happens if we replace n by infinity? Natural introduction of the little l_p spaces. Ex 1.1.3. What about function spaces, like C([a,b])? Ex 1.1.4,10. Abstract normed spaces. A norm gives a metric (distance concept), ex 1.1.5, sometimes a norm comes from an inner product, ex 1.1.6. What is special about the inner product norm? ex 1.1.8, real case hard enough. Which concepts and techniques rely on that? Why and what about complex spaces? Ex 1.1.7. Finite- vs infinite dimensional. Bases. Ex 1.3.1,2,3. CHAPTER 2 2.1 (why?) topology in metric spaces: metric -> definition of convergent sequence -> definition of sequentially compact sets metric -> open balls -> topology (def of open sets) -> definition of compactness In metric spaces: (Theorem 2.4) compact (difficult concept) <=> sequentially compact (easy concept) Proof uses notion of totally bounded sets. Likewise metric -> definition of convergent sequence -> definition of limit points -> definition of closed sets metric -> open balls -> open sets -> definition of closed sets Make a mental note to recognise the applications where topology offers an advantage not offered by what you can do with sequences. Heine-Borel in R^n: closed + bounded <=> sequentially compact. Heine-Borel does not hold in C([a,b]) and the little l_p spaces Ex 2.1.13 (a,b) In fact: Heine-Borel only holds in finite-dimensional normed spaces, Ex 2.1.13 (c) E in C([a,b]) is compact <=> E is closed, bounded AND equicontinuous In my memory the <= implication was stored as the Ascoli-Arzela Theorem. The => implication is discussed in relation to the sequence x^n, this sequence is easily seen (why?) not to have a convergent subsequence in C([0,1]). Caution: wrong definition of equicontinuity in book. Ex 2.1.1-6 are about topology in metric spaces (general math background) Ex 2.1.7,12 the discrete metric has no relevance in the context of normed spaces, why? Ex 2.1.9-15 are about compactness 2.2 Separability. If only countably many elements of X are needed to approximate every element in X arbitrarily close, X is called separable. Avoid non-separable spaces if possible. They are too large to handle. Ex 2.2.1-3. NB C([a,b]) is separable. Show this directly using the uniform continuity of functions in C([a,b]) using piecewise linear continuous functions with rational values in the rational begin- and endpoints. 2.3 Complete spaces. Cauchy sequences must be convergent for a (metric/normed) space to be usefull, spaces with this property are called complete. C([a,b]) with maximumnorm is complete, but not with respect to the inner product L^2-norm. Examples: Ex 2.3.5-8 (R is complete!) CHAPTER 3 (don't like this chapter very much) 3.1 is intended as motivation but in my view rather disconnected from what follows. each omega (w for short) in (0,1] can be uniquely written as a binary expansion, not (always) unique because 1/2 = 0.1000000000...... = 0.011111111111..... To make the expansion unique disallow tails with only zero's. Connection with coins by relating the n-th bit to head/tails in n-th coin toss. Convince yourself of the (nice) relation between probability of sets E of events (Bernouilli sequences) and the Lebesgue measure of the corresponding subsets B_E of (0,1] and make sure you understand both large number laws. Only partly covered by ex 3.1.1-4. 3.2 Lebesgue measure. This section is much too technical for my taste. I assume that most of you have seen this somewhere else. Lebesgue measure should follow in 6 steps i obvious definition for blocks (cells, intervals): length time width in R^2 etc. ii natural definition of outer measure with countable covers consisting of blocks (top page 38) iii outer measure is countable sub-additive, easy! iv observation that inner measure=outer measure idea does not work for definition of measurabel set. v the CLEVER idea introduction of the measurable sets as precisely those sets which cut any other set in equal parts. vi outer measure is countably additive on those measurable sets which form a sigma-ring (in short: countable operations reasonly expected to be true are true, only proof I would have to look up in my old notes from Jan van de Craats' course where the name sigma-algebra was used for sigma-ring). Plus the remark that you need only few properties of the collection of blocks. These properties can be axiomatised when/if needed. I had a hard time to tell the hard parts from the easy parts in this part of the book. Convince yourself or recall how open sets and Cantor set are obtained from countable set theoretic operations starting with intervals (and hence measurable in view of vi above). 3.3 Definition of measurable and summable functions. I assume that most of you have seen this somewhere else. 3.4 Limit theorems I assume that most of you have seen this somewhere else. 3.5 Lebesgue L^p spaces. I don't assume that most of you have seen this somewhere else. Introduction of equivalence classes of measurable functions L^p-norm (p=1, 1 all absolutely convergent series are convergent) L^p(R^n) is separable Real and complex cases. Ex 3.6.2-9 are relevant for standard applications. CHAPTER 4 Fourier series (only series) Construction of countable orthonormal bases in separable Hilbert spaces through Gramm-Schmidt. Fourierseries of periodic functions. Relation between the two, convergence in mean. Brief discussion of other types of convergence (examining Fourier series for f and F with F'=f). All exercises are about L^2. CHAPTER 5 Operators 5.1 matrices, integral operators (sum over index j replaced by integral over y) For linear maps: continuity in one point <=> continuity everywhere 5.2 bounded linear operators For linear maps: continuity in one point <=> continuity everywhere <=> bounded The norm estimate for T : X -> Y gives uniform Lipschitz continuity The best M is the norm of T Y Banach => B(X,Y) Banach (collection of all bounded linear T: X -> Y) 5.3 Banach algebra's and spectral theory. Y=X gives B(X)=B(X,X), not only a Banach space, but also an algebra, with unit element I. see my old notes fa.pdf, section 20. In algebra's, such as B(X), we can talk about invertibility. Basic tool for invertibility (I-T)^(-1)=I+T+T^2+T^3.... just the geometric series, convergent if T is not too large, e.g. norm below 1. T invertible and S close to T gives S invertible. Get used to the role of z complex being played by lambda. For asci reasons I now use z instead of lambda. For T in B(R^n) or (B(C^n) (i.e. T a matrix) the spectrum of T is the set of complex eigenvalues. Now we want to have complex spaces X, all examples naturally come with real and complex version hand in hand, but there is an abstract complexifciation procedure (norm definition is not obvious. Definition. z belongs to spectrum of (T) <=> z-T is NOT invertible in B(X) (everything complex now) The inverse defines a B(X)-valued function on the complement of the spectrum. This complement is called the resolvent set of T. The inverse of z-T is called the resolvent function (depends on A and z, for now A is fixed) Theorem. T in B(X) => sigma(T) closed (easy), bounded (easy) and NON-EMPTY (non-trivial) Proof: resolvent is complex analytic, goes to zero as z goes to infty. Liouville's theorem says that this is impossible for globally defined analytic functions. Wow! Small detail: B(X)-valued functions instead of C-valued functions. Book cheats a bit, details are postponed. Resolvent has power series expansion in 1/z. Convergent for |z| > ||T||. (In fact for |z| larger than spectral radius r(T)= lim ||T^n||^(1/n), not proved here) NB Real part of the spectrum may be empty. 5.3 continued, compact operators. In general sigma(T) consist of eigenvalues and many more other points. Special case, T is a compact operator T is compact <=> the image of every bounded sequence has a convergent subsequence => 0 is in sigma(T), all other point in sigma(T) are isolated eigenvalues, no accumulation points except possibly zero. K(X) = { compact bounded linear operators T: X -> X } K(X) is a closed ideal in B(X) 5.4 Invariant subspaces General question. Can we decompose T : X -> X into smaller parts? (think of a 4x4 matrix with zeros in the 2x2 squares top-right and bottom-left). Related question. Does T have an invariant subspace? (think of a 4x4 matrix with zeros in the 2x2 square bottom-left). Answer. In general no (counterexample hard), but yes if TK=KT for some compact bounded linear operator (non-zero of course, proof is hard/tricky). 5.4 back to concrete/easy cases, in the rest of the chapter X=H is a Hilbert space, T in B(H) is selfadjoint. Why complex? In Thm 5.18 we see that the spectrum is real. In fact the easiest theorem to prove follows from Lemma. Let H be a real Hilbert space, T: H -> H selfadjoint compact. The |(Tx,x)| has a maximum on the closed unit ball B AND every maximizer is an eigenvalue. Proof. Step 1. Maximum exists in view of compactness of T (a bit tricky, not hard). Step 2. Maximizers are eigenvectors (easy, not different from proof for 2x2 matrices). From Lemma a formulation and proof of spectral theorem follow easily. MORE TO FOLLOW