Lecture 1, September 13, 2004 We gave a 45 minute introduction to the field, indicating that functional analysis is a blend of linear algebra and topology (in our case: the theory of metric spaces). It is the theory of topological vector spaces and (continuous linear) maps between them. A historically important theorem on Sturm-Liouville problems was formulated, claiming the completeness of the eigenfunctions of rather general differential operators. The quest for such results has stimulated the development of the functional analytic approach, and we intend to understand this theorem by the end of the course. In the second part we set out on the preliminaries: linear algebra and metric spaces. We skipped the linear algebra part (read R&Y 1.1 if you want, but there is nothing new here), so the second part was entirely devoted to metric spaces. Basic notions were recalled at a quick pace and two new results were stated without proof: the Stone-Weierstrass theorem for compact metric spaces (R&Y states only a particular case) and the Baire category theorem on complete metric spaces. Read R&Y 1.2 but do not try to construct the proofs of all the statements yourself. Do the homework exercises in the first series instead: these should give you a good opportunity to work with the theory of metric spaces and continuous maps between them. If you can solve these, then your working knowledge of metric spaces is fine. Lecture 2, September 20, 2005 To finish the preliminaries, we gave an outline of the Lebesgue measure and integration theory. Be advised that the technique is a bit more involved than the outline perhaps suggested, and that this was not always mentioned - so don't rely on these sketchy notes if your life depends on it! Read R&Y 1.3 to complete the picture, but do not try to understand all the details. The moral is that there exists a theory of integration which is built on a theory of measure and which is much more general and stronger than the theory of the Riemann integral, containing the latter as a special case. It has all the properties which one would expect from one's knowledge of the Riemann integral. One can adapt the strategy of mentally substituting a Riemann integral of a piecewise continuous function over a finite interval whenever a Lebesgue integral is encountered. This is not correct of course, but the risk of damage is limited. Next we introduced the concept of normed spaces and gave some examples: spaces of continuous functions, finite dimensional spaces with p-norms and the L_p-spaces which one constructs in the Lebesgue theory. For the L_p-spaces one can pretend that an equivalence class of functions, which occurs in the construction, is a continuous function on a bounded interval and that the norm is simply calculated as a Riemann integral. This is again not correct, but it helps. (Remark: the canonical map of C[a,b] into L_p[a,b] is an injection which is, however, by no means surjective. L_p[a,b] is much larger than C[a,b]. If p is finite, then L_p[a,b] is a model for the completion of C[a,b] under the metric which comes from the p-norm. This follows from the density result Theorem 1.61 in R&Y, which is not at all trivial). As a special case of the general L_p-spaces we obtained the sequence spaces l_p. Read R&Y 2.1 up to and including Example 2.6. The L_p-spaces are very important normed spaces, but their definition is admittedly a bit technical. It seems that we are through with that now and that we can proceed with the general theory next time. Lecture 3, September 27, 2004 We finished Chapter 2 in R&Y: If X is a normed space, then it has an associated metric topology, so that now finally the combination of the two basic ingredients of FA (linear structure and topology) has become visible. One can build new normed spaces from old ones, by considering subspaces, products and quotients by closed subspaces (the latter is not in R&Y). The corresponding norm topologies are then precisely the ones which result from the general constructions in topology. Two norms are equivalent (as defined in R&Y) precisely when their associated topologies are the same. This result is not in R&Y (but it is in the homework). We treated the basic material on finite dimensional normed spaces: all norms are equivalent and the space is always complete. If a subspace of a normed space is finite dimensional, then it is closed. Finally, we stated the important Riesz Lemma (R&Y 2.25), which is used in proving that the unit ball or sphere is compact precisely when the dimension is finite. This is a topological characterization of finite dimensionality of normed spaces (note the blend of topology and linear algebra!). Read the rest of the second chapter in R&Y, starting with Example 2.7 and have a look at the exercises and their solutions. Note exercise 2.12 on page 50, demonstrating that the closure of an open ball in a normed space is the closed ball with the same center and radius (cf. a homework exercise in the first series: this is not true for arbitrary metric spaces). Lecture 4, October 11, 2004 We did the paragraphs R&Y 3.1, 3.2 ane 3.3. Starting with the basis definitions of an inner product space, the Cauchy-Schwarz inequality and the associated norm, we defined he notion of a Hilbert space. "The only" example of a Hilbert space, namely L_2(X), was explained. R^n, C^n, l_2, and of course L_2[0,1], are all particular cases of this universal model (which was also a good excuse to mention the Hoelder inequalities from R&Y 1.54). Subspaces and products of inner product spaces carry a natural inner product. A particular feature of a norm which comes from an inner product is the parallellogram rule which it satisfies. Although we did not prove this, it is in fact true that a norm which satisfies this rule comes from an inner product. How one should define this product is made obvious by the so-called polarization identities, expressing an inner product in terms of norms. One has the usual notions of orthogonality, orhonormality and the Gram-Schmidt procedure, familiar form linear algebra. Things start to get interesting in the geometrical Theorem 3.32, stating that the distance between a closed convex subset of a Hilbert space and a point is always realized by a unique point of the subset. This is the basis for the workhorse of the subject, Theorem 3.34, which shows that, given a closed subspace Y of a Hilbert space, the whole space has an orthogonal decomposition into Y and its orthoplement. Such a decomposition result does not hold for general Banach spaces: there are examples of a Banach space X and a closed subspace Y where there does not exist a closed subspace Z such that X is the direct sum of Y and Z. Closely related to the decomposition theorem is the description in Corollary 3.36 of the double orthoplement of any subspace of a Hilbert space: you simply get the closure of what you started with. Read R&Y 3.1 - 3.3. The exercises don't seem to be particularly challenging, with perhaps 3.20 on page 72 as an exception. Part (a) shows that Corollary 3.36 holds for arbitrary inner product spaces and part (b) is a logical test. Lecture 5, October 25, 2004 We did an extended version of R&Y 3.4 and 3.5. Any Hilbert space has an orthonormal basis (a consequence of Zorn's Lemma) and the cardinality of such a basis is well-defined, thus determining the isomorphism class of the Hilbert space. The Hilbert spaces with a countable orthonormal basis are precisely the separable ones. Each Hilbert space is isomorphic to an L_2-space for a counting measure. An element of a Hilbert space can be developed as a generalized Fourier series in terms of an orthonormal basis, with only countably many terms being nonzero and the sum of the series being independent of the ordering of the terms. For L_2-spaces on the real line one has examples of orthonormal bases consisting of complex exponentials, sines or cosines. Lecture 6, November 1, 2004 We covered the very basics of (bounded linear) operators between normed spaces. If X and Y are normed spaces, then so is B(X,Y); furthermore, if Y is Banach, then so is again B(X,Y). If X is Banach, then B(X) is an example of a Banach algebra (more about this in next semester's seminar!) where multiplication is given by composition of maps. A particularly important example is B(X,F)=X', the dual space. It is convenient to know concrete models for dual spaces. If p>1 is finite, then the dual of L_p can be identified with L_q through the natural pairing. The same holds for p=1 if the measure is sigma-finite. Hilbert spaces can be identified (in an antilinear fashion) with their own dual in a natural way (Riesz-Frechet). Undoubtedly the most important theorem in this field is THE Ries Representation Theorem, stating that the dual of C_0(X), i.e., of the space of functions vanishing at infinity on a locally compact Hausdorff space (supplied with the sup-norm), can through the natural pairing be identified with the space of regular Borel measures on X (with the total variation as norm). See Sander Hille's course for more material on this. Read 4.1-4.3 for the above material and some complements. In addition, we covered some material which is not in the book for some odd reason. To start with, we stated the truly fundamental Hahn-Banach theorem: if X is a normed space and if L is a subspace, then any element of L' can be extended in at least one fashion to an element of X' with the same norm. As a consequence, X' separates the points in X and X' is always nontrivial if X is nontrivial. As another consequence, X has a canonical and isometric embedding into its second dual X'', which gives an easy way of showing that the metric completion of a normed space is a Banach space. If the embedding is surjective, then X is called reflexive. Hilbert spaces are reflexive, as are all L_p-spaces for finite p>1. Lecture 7, November 8, 204 Continuing basic operator theory, we covered 4.4 in a slighly extended version. If X is a Banach space, then the group of invertible elements in B(X) is an open and topological subgroup of B(X). Have a look at Example 4.41 (we did it in greater generality) about the existence of a solution to an integral equation. This is the kind of example which stimulated the development of functional analysis in its early days: one can not see this with "elementary" analysis as in the first two undergraduate years, one really needs a higher level of abstraction. The open mapping theorem states that a surjection between Banach spaces is an open map. Its twin brother is the closed graph theorem, stating that a linear map between Banach spaces is continuous precisely when its graph is closed. The uniform boundedness principle (hidden in exercise 4.22) is one of the other basic theorems in functional analysis: if a collection of bounded linear maps from a Banach space into normed linear spaces is poinwise bounded, then it is uniformly bounded. It is also known as the Banach-Steinhaus theorem. Invertibility in infinite dimension is a delicate matter. Theorem 4.48 gives necessary and sufficient conditions which are important in concrete situations. We also covered the beginning of Chapter 5 (roughly up to and including Theorem 5.10), entering the well-developed and historically important field of operator theory on Hilbert spaces. The definition of the Hilbert space adjoint is as in linear algebra and many basic properties are as you might expect, but the statements about operator norms in Theorem 5.10 may be new to you. If A is a subalgebra of B(H) which is closed and selfadjoint (i.e., invariant under the adjoint map), then A has all the definining properties of a C*-algebra. These algebras have become rather important in contemporary mathematics and physics. The not-at-all-trivial Gelfand-Naimark theorem states that any abstract C*-algebra is in fact isomorphic to a closed selfadjoint subalgebra of B(H) for some suitable Hilbert space H. More about C*-algebras in next semester's seminar. Lecture 8, November 15, 2004 We continued the study of operators on Hilbert spaces (and on Banach spaces), covering 5.11 through 5.41 (but not yet the numerical range in 5.41). The main points were the introduction of normal operators (with self-adjoint and unitary operators as special cases) and of the spectrum of an operator. The spectrum is the generalization of the set of eigenvalues in linear algebra. For an operator T on a complex Banach space, the spectrum is non-empty and compact, and it is contained in the disc with 0 as center and ||T|| as radius. We mentioned (but did not prove) that the maximum modulus of the numbers in the spectrum is given by the limit (which exists), as n tends to infinity, of the n-th root of ||T^n||. This is the spectral radius formula, which is true in general complex Banach algebras. If T is normal, then the spectrum and the approximate point spectrum coincide. Later on we will see how this implies that every nonzero number in the spectrum is in fact an eigenvalue if T is compact and normal, implying that T has an orthonormal basis of eigenvectors. This spectral theorem is at the heart of many completeness results in analysis and we will see later on how it implies the completeness result for Sturm-Liouville theorems as in the beginning of the course. If p is a polynomial, then the spectrum of p(T) is easily determined from that of T, with a similar result for the spectrum of the inverse of T (if it exists) - see the noteworthy Theorem 5.39 for this. As a sidestep, we mentioned that the normality of T is equivalent to the commutativity of the C*-algebra generated by T. The commutative Gelfand-Naimark states that any commutative C*-algebra with 1 is isomorphic to a C*-algebra C(X) for a suitable compact Hausdorff space X. In case of a normal operator T we have a very concrete model: the C*-algebra generated by T is isomorphic to the C*-algebra of all continuous functions on the spectrum of T, under an isomorphism which sends T to the function id, where id(z)=z. Many results for normal operators - which we will prove "by hand" in this course - follow almost immediately from this isomorphism. Lecture 9, November 22, 2004 We started with the definition of the numerical range in 5.41 and finished Chapter 5. Theorem 5.43 is noteworthy, especially b and d. Part d is also true for normal operators in general. We split 5.4 in two parts, 5.4.a on positive operators and 5.4.b on projections. As to the positive operators, we upgraded Lemma 5.57 to a Theorem stating that there is an isometric embedding of the algebra of real valued continuous functions on the spectrum of a self-adjoint operator into B(H), and which is the obvious mapon R[X]. The image is contained in the self-adjoint operators. This makes the existence of square roots a triviality as in 5.58, the uniqueness still requires an argumentation (as also in 5.58). From the existence of a square root it is only a short step to the polar decomposition of an invertible operator in 5.59. We also stated the general polar decomposition for not necessarily invertible operators. Partial isometries show up here. As to the projections, the fundamental observation to be made is that giving an orthogonal projection (i.e., a self-adjoint idempotent) is the same thing as giving its range: the orthogonal projection P is the canonical projection onto Im P which is associated to the splitting of H as the direct sum of Im P and the orthoplement of Im P. This explains the adjective "orthogonal" in the name. In the partial ordering one has P<=Q if and only if Im P is contained in Im Q, so that the partially ordered set of projections is isomorphic (as a partially ordered set) with the partially ordered set of all closed linear subspaces of H. As a consequence of this isomorphism, each set of projections has a supremum and an infimum in the partially ordered set of projections and the isomorphisms tells us what these are. Lecture 10, November 29, 2004 We covered the basics of compact operators and began with the spectral theory. The basics are described in R&Y section 6.1. One of the main results is that K(X) (where X is a Banach space) is a closed two-sided ideal in B(X). The quotient B(X)/K(X) is then not only a normed space, but also an algebra and it is in fact a Banach algebra. This so-called Calkin algebra is important in operator theory. A famous result of Enflo from 1973 is an example of a separable reflexive Banach space and a compact operator on it which is not the limit of a sequence of finite rank operator. However, a compact operators between a normed space and a Hilbert space is always such a limit (R&Y 6.12). It is in general true that an operator between Banach spaces is compact if and only if its tranpose (between the duals) is compact. R&Y 6.14 is a restatement of this result in the Hilbert space context, with an approach using finite rank operators. We will not use Hilbert-Schmidt operators (6.15 and 6.16). R&Y will probably want to use this to show compactness of integral operators in the last chapter, but we will base ourselves on a more general approach using the Arzela-Ascoli theorem. The spectral theory for compact operators in R&Y 6.2 is developed for Hilbert spaces. While this context is technically convenient, it obscures the fact that the results of this paragraph are actually valid for complex Banach spaces, provided that one replaces the adjoint of an operator in R&Y with its tranpose in the general case. E.g., the spectrum of a compact operator T on an infinite dimensional complex Banach space contains zero and the non-zero elements in the spectrum are either finite in number, or else form a sequence which converges to zero. Furthermore, each non-zero element of the spectrum is an eigenvalue with finite multiplicity and one has dim Ker (s-T) = codim Im (s-T) for all non-zero s. Im (s-T) is closed for non-zero s. We will say even a bit more about these operators which has an interpretation in terms of Jordan blocks to complete the picture, but most likely without proof. Lecture 11, December 6, 2004 We finished the spectral theory of a compact operator T on a complex Banach space X. In addition to the results mentioned in the description of the previous lecture, we mentioned (but did not prove) that the dimension of Ker (s-T) is equal to the codimension of Im (s-T) if s is nonzero, and that they are also equal to the same numbers for T' (the transpose of T). In particular, for nonzero s the surjectivity of s-T and the injectivity of s-T are equivalent (compare with finite dimensional linear algebra!). This so-called Fredholm alternative is perhaps more easily remembered in this form than as it is stated in many books, e.g. in R&Y 6.26. Furthermore, the descending chain of Im (s-T)^n and the ascending chain of Ker (s-T)^n both stabilize, and at the same spot. If this spot is k, then X is the direct sum of the Ker (s-T)^k and Im (s-T)^k, where both closed subspaces are invariant under T (compare with the Fitting decomposition in finite dimensional linear algebra!). If s is in the spectrum of T, then the spectrum of T on the first space is just s, and on the second space it is the spectrum of T on X with s removed. The restriction of T on the first space can be described by a finite number of Jordan blocks of finite length. We did not pay attention to R&Y 6.29. The spectral theory for compact normal operators on a Hilbert space is easily summarized: these can be diagonalized with respect to an orthonormal basis. We did not cover R&Y 6.36 and 6.37. Lecture 12, December 13, 2004 We covered the material in 7.1 and 7.2 on integral operators. The Fredholm operators have continuous kernel on the square, whereas the Volterra operators have continuous kernel on the right lower triangle in the square and are zero on the left upper triangle. Both define compact operators from L_2[a,b] into C[a,b], and hence also compact operators from C[a,b] into intself and from L_2[a,b] into itself. The basis for this compactness is the Ascoli-Arzela theorem (R&Y use Hilbert-Schmidt theory for this, but whereas this shows compactness of the operator on L_2[a,b] for arbitrary square integrable kernel, it does not give compactness of the operator on C[a,b]). The spectrum of a Fredholm integral operator on C[a,b] is the same as that on L_2[a,b], as are the eigenfunctions corresponding to non-zero eigenvalues. If the kernel of a Fredholm integral operator is hermitian, then it yields an orthonormal bases of eigenfunctions of L_2[a,b]. The spectrum of a Volterra integral operator consists of just 0, both on C[a,b] and on L_2[a,b]. We did not pay attention to 7.8-7.12. Lecture 13, December 20, 2004 We skipped section 7.3 in R&Y. In section 7.4 on Sturm-Liouville theory, the basic idea is to invert the operator Lu=u''+qu with an integral operator. More precise: if C^2_0 is the space of all C^2-functions vanishing at a and b, and if L has trivial kernel on C^2_0, then L is a bijection between C^2_0 and C[a,b] with a Fredholm integral operator G as inverse. The kernel of this operator (Green's function) is continuous and hermitian (in fact symmetric), so that the spectral theory for self adjoint compact operators comes within reach. The definition of this kernel (R&Y 7.24) comes admittedly out of the blue, but it can more satisfactorily also be derived using variation of constants (which we did not do, however, in view of the time). If L has trivial kernel, then the eigenfunctions of L in C^2_0 are precisely those of G (with the reciprocal eigenvalues), so that the eigenfunctions of L can be normalized to form an orthonormal basis in L_2, in view of the spectral theorem for G. The condition that Ker L is trivial is in fact redundant for this to be true, because one can always replace L with L-s, with s real and such that L - s has trivial kernel (we proved that such s exists). So, if f is in L_2, then f can be developed as a Fourier series in terms of the eigenfunctions of L in C^2_0. If f is in C^2_0, then we have a much stronger result: this Fourier series converges to f even in the norm |.|_1, where |f|_1=|f|_\infty + |f'|_\infty (this is a stronger statement than R&Y 7.28). To summarize: Let q in C[a,b] be real-valued. Then the set of complex numbers s such that the equation u''+qu=su has a non-trivial solution in C^2_0[a,b] is countable, real, and can be ordered as a sequence s_1>s_2>s_3>... which diverges to minus infinity. The corresponding solution spaces are one-dimensional and yield an orthonormal basis of L_2[a,b], so that any f in L_2[a,b] can be developed as a Fourier series in terms of the eigenfunctions. If f is in C^2_0[a,b], then this Fourier series is in fact convergent to f in the norm |.|_1 defined above. The Sturm-Liouville theory as we have developed it has many extensions to other second order operators (in a sende: to all, provided that the coefficients are sufficiently regular), to other boundary conditions and to higher order operators. Similar results on orthonormal bases consisting of eigenfunctions of differential operators exists in higher dimensions. The technique is then more involved, but the heart of the proof remains an application of the spectral theorem for compact self-adjoint operators on a Hilbert space.