A Course in Analysis

The following sections are included:

• Preface
• Introduction
• Contents
• List of Symbols

This chapter is not an introduction to the book, nor is it meant as a starting point for a reader who starts studying functional analysis. It is intended to inform a reader with some background in functional analysis, for example someone who has mastered Chapters 2-9, from a more advanced point of view about our approach to the topic. In particular, we believe that lecturers using this text in their course may benefit from this chapter…

Vector spaces we encountered rather early in our Course and they are the most common algebraic structure beyond ℝ and ℂ we worked with so far. The family of vector spaces we have been working with is quite diverse, for example we used the finite dimensional spaces ℝⁿ or ℂⁿ, spaces of sequences such as l_p, or spaces of functions, e.g. C(G) or C^∞(G), and even spaces of equivalence classes of functions, namely L^p(G). In some situations, we also looked at vector spaces of linear mappings, for example the vector spaces M(n,m;ℝ). Readers are assumed to be able to work with vector spaces as in (linear) algebra or geometry. In Appendix I of Volume II, we have summarized basic, maybe also some advanced material from linear algebra, i.e. the theory of finite dimensional vector spaces. In functional analysis, we are interested in infinite dimensional vector spaces which we can equip with a topology compatible with the algebraic structure. Before we can study the relations between the algebraic (linear) and topological structure we need a better understanding of the algebraic structure itself. We will always work with 𝕂-vector spaces, 𝕂 ∈ {ℝ,ℂ}. In particular we need a clarification of the notion “algebraic basis”. The central existence result, i.e. the statement that every vector space has a basis, will depend on the axiom of choice or, more precisely, one of its equivalent formulations, namely the Lemma of Zorn. The reader is reminded that we have used Zorn’s lemma already in Volume III when discussing the existence of a subset of [0, 1] which is not Lebesgue measurable, see Appendix II in Volume III…

The reader is assumed to have basic knowledge of norms and normed spaces as developed in previous volumes, especially Volume II. In particular we take for granted that the reader knows that every norm induces a metric which in turn gives rise to a topology. Thus the notions of convergence with respect to a norm and Cauchy sequence with respect to a norm as well as that of continuity of mappings between two normed spaces we take for granted. Some properties of norms we recollect in Problem 2. We recall some basic properties of sequences and Cauchy sequences…

For two 𝕂-vector spaces X and Y, we define as usual a mapping T : X → Y as linear if for all x₁, x₂ ∈ X and λ, μ ∈ 𝕂

We recall

Definition 5.1. The dual space X* of a normed 𝕂-vector space (X, ⃦ · ⃦) is the space L(X, 𝕂), 𝕂 ∈ {ℝ,ℂ}.

We know that X* is with respect to the operator norm always a Banach space, see Proposition 4.13. It turns out that with the help of X* we can often understand the space X much better, most of all X* is a very powerful tool to solve equations. Before investigating X* we want to “identify” X* for several well-known Banach spaces X with a known Banach space Y…

Many far reaching results in functional analysis and its applications rest on a few very central results which we call the “basic principles”. They are related to a result which is of topological nature and which we discuss first…

The basic goal of functional analysis is to study operator equations. For a (linear) operator T : D(T) → Y defined on some domain D(T) ⊂ X where X and Y are (at the moment) vector spaces, we seek to find all x ∈ D(T) such that Tx = y holds where y ∈ Y is given. Even in the case of finite dimensional vector spaces we know that a solution need not exist or that a solution need not be unique. Thus we are longing for solvability conditions as well as a characterization of the solution manifold. It turns out that many questions related to the solvability problem for an operator equation in normed spaces are best understood when employing the dual spaces. As a first step we will investigate adjoint operators of linear operators T : X → Y…

By definition, a Hilbert space is a scalar product space which is complete with respect to the norm induced by the scalar product. We have encountered scalar product spaces (Appendix IV.A.I) and Hilbert spaces (Chapter IV.5 and Chapter IV.9) before and in the first part of this chapter we summarize the main results. In only a few cases it is of some advantage to handle Hilbert spaces over ℝ and then most results follow in a natural way from those for Hilbert spaces over ℂ. Therefore, unless stated otherwise we will assume in this chapter that 𝕂 = ℂ…

We have seen that certain operators of interest are unbounded, e.g. differential operators or inverse operators of injective compact operator defined on the range of the original operator. However, defined on proper domains these operators are closed or at least closable, we refer to Definition 6.24 and the following Definition 9.1. In this chapter we want to study unbounded but closed or closable linear operators in separable Hilbert spaces. In particular, we shall investigate self-adjoint operators and their spectral theory. Let us first summarise some definitions…

The spectral theorem for self-adjoint compact operators on a Hilbert space, Theorem 8.47, has already led to many further interesting theoretical considerations as well as applications to concrete operator equations, e.g. certain integral equations. A natural question is whether we can extend this theorem to bounded or even unbounded self-adjoint operators, or to non-self-adjoint operators. The case of bounded operators is best covered by the theory of Banach algebras which we will study with the application to spectral theory in mind. The Cayley transform as discussed in the last chapter will then allow us to prove a spectral theorem for unbounded self-adjoint Hilbert space operators in the following chapter…

We want to investigate bounded and unbounded self-adjoint operators T in a separable Hilbert space (H, ❬·, ·❭) over ℂ. Let T ∈ L(H) be a bounded self-adjoint operator and λ ∈ ρ(T), i.e. λ belongs to the resolvent set of T. By Proposition 10.6, the resolvent set ρ(T) ∈ ℂ is open and for λ₀ ∈ ρ(T) there exists δ > 0 such that in D_δ (λ₀) = {λ ∈ ℂ | |λ−λ₀| < δ} the resolvent R : ρ(T) → L(H) admits a power series representation, i.e.

So far we have studied mainly Banach spaces and Hilbert spaces as well as (linear) operators between (subsets of) such spaces. However, in some situations we had to deal with function spaces which are not Banach spaces but still admit a notion of convergence or a topology compatible with the vector space structure. As an example we may consider the space of all continuous functions defined on an open set G ⊂ ℝⁿ and the uniform convergence on compact subsets. With this notion of convergence C(G) is complete, but it is not a normed space. A further result in this direction is Montel’s Theorem, Theorem III.28.13…

We have discussed convex subsets of ℝⁿ in Chapter II.13 and as some of the central results we mention

• the existence of the metric projection onto a closed convex set K ⊂ ℝⁿ, Theorem II.13.14, a result which we could extend to Hilbert spaces, Theorem 8.16;
• the Minkowski theorem, Theorem II.13.24, stating for a compact convex set K ⊂ ℝⁿ that conv(extK) = K where conv(A) is the convex hull of A ⊂ ℝⁿ and extK is the set of all extremal points of K;
• (continuous) convex functions f : K → ℝⁿ on a compact convex set K ⊂ ℝⁿ attain their maximum in an extreme point, Theorem II.13.25;
• convex functions on an open convex set are continuous, Theorem II.13.27, in fact they are locally Lipschitz continuous, hence in one dimension they are almost everywhere differentiable.

In this chapter we want to discuss two separate topics, however each is worth a monograph alone. We will start with the classical M. Riesz-Thorin theorem in interpolation theory and we will follow the subject until the theorem characterizing the domains of fractional powers of positive definite self-adjoint Hilbert space operators as complex interpolation spaces. We then turn to real interpolation and prove the Theorem of Marcinkiewicz. The second topic, the Lax-Milgram theorem, is first considered as a further representation theorem for linear functionals on Hilbert spaces and then applied to sesquilinear forms satisfying an abstract Gårding inequality…

We want to study some classes of integral operators

While integral operators are natural objects to study within a first course on functional analysis, to some extent investigations on integral equations initiated the subject, operator semigroups are often treated only rather briefly. We find that there are several good reasons to study operator semigroups early: they have plenty of applications in such diverse fields such as partial differential equations, spectral theory, the theory of Markov processes, mathematical biology, control theory, just to name some, but in addition they force us to study unbounded operators in a natural context and to look more closely at the interaction of (abstract) functional analysis and (concrete) operators such as partial differential operators…

Banach spaces consisting of complex-valued functions u : G → ℂ contain the cone of non-negative functions and in some of the most interesting examples the order structure induced by the cone of non-negative functions is related to the norm. For example if u, v ∈ L^p(G, μ) and 0 ≤ u ≤ v μ-almost everywhere then ${\Vert u\Vert }_{Lp}\le {\Vert v\Vert }_{Lp}$. It is natural to study linear operators in such Banach function spaces which preserve the order, i.e. operators with the property that u ≤ v implies Su ≤ Sv. Instead of introducing a new “general structure” such as a Banach lattice, see [129] or [111], we restrict ourselves here to the case of spaces of continuous functions, (Borel-) measurable functions, or the L^p-spaces. On the one hand side, operators preserving order relations in these spaces are best studied in Banach spaces over ℝ. However, certain aspects of the general theory of linear operators in Banach spaces, e.g. spectral theory, we prefer to study in Banach spaces over ℂ. The following schemes of complexification of a real Banach space and a linear operator on a real Banach space show a possibility to combine both…

In Chapter IV.9 we discussed orthonormal expansions with respect to special functions in some L²-spaces, and in Chapter IV.21 as well as in Appendix IV.A.III we could relate these special functions to certain second order linear differential operators, e.g. Sturm-Liouville operators. More precisely, these functions were eigenfunctions of these operators and it is natural to ask how the expansions previously discussed relate to the spectral theorem. As we have seen, differential operators are unbounded operators in L²-spaces. While the expansion results mentioned above remind more on the spectral theorem for symmetric compact operators in a separable Hilbert space, a direct application of this theorem to differential operators is not possible. In this chapter we want to clarify the situation and we will find the appropriate context to handle regular Sturm-Liouville operators in the frame of (abstract) functional analysis…

Roughly speaking, Sobolev spaces are spaces of L^p-functions defined on some measurable subsets of ℝⁿ which admit certain generalized partial derivatives belonging to L^p too. Equipped with a natural norm they are Banach spaces (for p = 2 they are Hilbert spaces) and as such they could serve as further interesting Banach space and hence might have a natural space in Part 12. They are also spaces of distributions, in fact we can identify generalized derivatives with distributional derivatives and this would give Sobolev spaces a canonical place in Part 14. To place a treatment of Sobolev spaces in the part being devoted to operator theory might look unnatural. However, we consider a core topic of operator theory to be the provision of a framework for handling (linear) operator equations in infinite dimensional spaces using tools from functional analysis. Tools such as the Hahn-Banach theorem or the Lax-Migram theorem have already been seen at work and spectral theory taught us that in many interesting cases we have to consider closures of operators and hence an investigation of the domain of such closures becomes a natural part of operator theory. Classical Sobolev space grew out of the discussion of closures of certain (partial) differential operators, partly in connection with the calculus of variations and for these reasons Sobolev spaces have their place in operator theory too. We want to start our considerations with a lengthy introductory example…

The classical Dirichlet problem for the Laplace operator

This chapter is devoted to three independent topics, Stone’s theorem on generators of strongly unitary operator groups, some perturbation results for self-adjointness, and finally we will discuss some first ideas related to the representation of groups by linear operators…

We want to implement locally convex topologies on certain vector spaces of differential functions which will turn these spaces into Fréchet spaces. We need…

Since the first third of the 20^th century it became apparent that classical analysis cannot provide sufficient tools to handle many problems in mathematical physics or other areas of applied mathematics. Singular behaviour of solutions of partial differential equations or even the proof of the Fourier inversion theorem required a different type of analysis. It was also noted that the way of dealing with singularities as in complex variable theory cannot in general resolve the situation. Dirac’s delta functions and Sobolev’s generalized functions emerged, on the more practical side Heaviside’s operational calculus introduced new ideas. J. Lützen [101] gave a very readable account on the history of these developments which finally led L. Schwartz to introduce “distribution” in [137] and [138]. In this chapter we start to develop the theory of distributions as it is nowadays widespread used in analysis. We follow closely our presentation [74] which is based on L. Hörmander [69] and [70], G. Friedlander [51] and C. Zuily [171]. Further readable introductions are those of J. J. Duistermaat and J. Kolk [38] as well as G. Grubb [62]…

In this chapter we study distributions in more detail, in particular we want to investigate distributions with compact support, the convolution of distributions and their singular support. We follow mainly our monograph [74] which, as mentioned before, is relying on the monographs of [70], [51] and partly [127] and [158]…

Distributions u ∈ 𝒟′ (ℝⁿ) which admit a continuous extension from 𝒟(ℝⁿ) to 𝒮(ℝⁿ) form a very powerful tool in analysis. They allow us to extend the Fourier transform from 𝒮(ℝⁿ) to this class of distributions and this will lead to many far reaching results. We start our investigations with…

The main result in this chapter is the kernel theorem due to L. Schwartz which characterizes a sequentially continuous linear operator from 𝒟(G₂) to 𝒟′ (G₁), G_j ⊂ ℝ^n_j open, with the help of a distribution K ∈ 𝒟′ (G₁ × G₂) called the kernel of the operator. It is clear, that once the existence of the kernel K is established, properties of the operator can be reduced to those of the kernel. In order to formulate and to derive the kernel theorem we first have to investigate tensor products. It is possible to discuss these topics more in the context of topological vector spaces, see for example [158], but we prefer more to follow the “hard analysis” approach and our main source is [70] as we have used his presentation in [74]…

In order to get an idea of the type of operators we want to study we have a closer look at the Hilbert transform, see (20.33) or the simplified version of it, i.e. the operator

The following sections are included:

• Appendix I: Completeness
• Appendix II: Nets, Convergence and Continuity
• Appendix III: On the Riesz Representation Theorem

The following sections are included:

• References
• Mathematicians Contributing to Analysis (Continued)
• Subject Index