Topics in Functional Analysis

In this chapter, we continue our study of Banach and Hilbert spaces. Here, we mainly consider linear functionals, i.e., additive, homogeneous, and continuous number functions given on such spaces. The problems under consideration are mostly grouped around two fundamental facts, namely, the Hahn-Banach theorem on extensions of linear functionals and the Banach-Steinhaus theorem (the principle of uniform boundedness). The general form of linear continuous functionals in many important spaces and some geometric problems in the theory of Hilbert spaces are also investigated.

Let K be a compact Hausdorff space and let )~ be a probability measure on K. We denote by Lo(K,2) the space of all Borel functions f:K--*N with the topology of convergence in measure. Lo(K, 2) is an F-space (complete metric topological vector space) if, as usual, we identify functions equal almost everywhere. Spaces of the type Lo(K,2 ) are probably the most studied examples of non-locally convex topological vector spaces. In spite of their bad reputation there is some evidence that they are in fact rather well-behaved spaces. Thus for a general topological vector space X it is often very useful to be able to produce nontrivial linear operators T:X--*Lo(K,2 ), in the same way as linear functionals facilitate the theory of locally convex spaces. We shall say that a point xeX is pathological it wherever T:X-*Lo(K, 2) is a continuous linear operator then Tx=O. X is pathological if every xsX is pathological. Note that if X is separable then it suffices in the definition to take K =(0, 1) with Lebesgue measure. The first example of a pathological F-space was given in 1973 by Christensen and Herer [3]. A more natural example is the space Lp/Hp where 0 < p < 1 (see [1, 5]). However the Christensen-Herer example also showed the connection with pathological submeasures. We recall that if d is an algebra of subsets of some abstract set L then a submeasure 4~:d--,lR is a map satisfying ~b(A)<_~b(AwB)<d~(A)+O(B) and th(O) = 0. 4~ is said to be pathological (see [3, 11]) if whenever 2 : d ~ N is a (finitelyadditive) measure with 0 <)~(A) < th(A) (At d ) then )~0. Now if S(~) denotes the space of all simple d-measurable functions f:L~IR, then S(~) can be topologized by the topology of convergence in @measure (functions differing only on a set of @measure zero are identified). The completion of S(~) in this topology may be denoted by A(~b). Then A(q~) is pathological

Application of neustadt's theory of extremals to an optimal control problem with a functional differential equation and a functional inequality constraint

Let ϕ : X → Y be an affine continuous surjection between compact convex sets. Suppose that the canonical copy of the space of real-valued affine continuous functions on Y in the space of real-valued affine continuous functions on X is complemented. We show that if F is a topological vector space, then f : Y → F is of affine Baire class α whenever the composition f ◦ ϕ is of affine Baire class α. This abstract result is applied to extend known results on affine Baire classes of strongly affine Baire mappings.

Instrumental variable methods for recovering continuous linear functionals

This chapter takes up several independent topics in Euclidean Fourier analysis, all having some bearing on the subject of partial differential equations. Section 1 elaborates on the relationship between the Fourier transform and the Schwartz space, the subspace of $L^{1}(\mathbb{R}^{N})$ consisting of smooth functions with the property that the product of any iterated partial derivative of the function with any polynomial is bounded. It is possible to make the Schwartz space into a metric space, and then one can consider the space of continuous linear functionals; these continuous linear functionals are called “tempered distributions.” The Fourier transform carries the space of tempered distributions in one-one fashion onto itself. Section 2 concerns weak derivatives, and the main result is Sobolev’s Theorem, which tells how to recover information about ordinary derivatives from information about weak derivatives. Weak derivatives are easy to manipulate, and Sobolev’s Theorem is therefore a helpful tool for handling derivatives without continually having to check the validity of interchanges of limits. Sections 3–4 concern harmonic functions, those functions on open sets in Euclidean space that are annihilated by the Laplacian. The main results of Section 3 are a characterization of harmonic functions in terms of a mean-value property, a reflection principle that allows the extension to all of Euclidean space of any harmonic function in a half space that vanishes at the boundary, and a result of Liouville that the only bounded harmonic functions in all of Euclidean space are the constants. The main result of Section 4 is a converse to properties of Poisson integrals for half spaces, showing that harmonic functions in a half space are given as Poisson integrals of functions or of finite complex measures if their $L^{p}$ norms over translates of the bounding Euclidean space are bounded. Sections 5–6 concern the Calderon–Zygmund Theorem, a far-reaching generalization of the theorem concerning the boundedness of the Hilbert transform. Section 5 gives the statement and proof, and two applications are the subject of Section 6. One of the applications is to Riesz transforms, and the other is to the Beltrami equation, whose solutions are “quasiconformal mappings.” Sections 7–8 concern multiple Fourier series for smooth periodic functions. The theory is established in Section 7, and an application to traces of integral operators is given in Section 8.

The mutational equations of Aubin extend ordinary differential equations to metric spaces (with compact balls). In first-order geometric evolutions, however, the topological boundary need not be continuous in the sense of Painleve–Kuratowski. So this paper suggests a generalization of Aubin’s mutational equations that extends classical notions of dynamical systems and functional analysis beyond the traditional border of vector spaces: Distribution– like solutions are introduced in a set just supplied with a countable family of (possibly non-symmetric) distance functions. Moreover their existence is proved by means of Euler approximations and a form of “weak” sequential compactness (although no continuous linear forms are available beyond topological vector spaces). This general framework is applied to a first-order geometric example, i.e. compact subsets of the Euclidean space evolving according to the nonlocal properties of both the current set and its proximal normal cones. Here neither regularity assumptions about the boundaries nor the inclusion principle are required. In particular, we specify sufficient conditions for the uniqueness of these solutions.

In this chapter the space of distributions is introduced and studied from the perspective of a topological vector space with various other additional features, such as the concept of support, multiplication with a smooth function, distributional derivatives, tensor product, and a partially defined convolution product. Here the nature of distributions with higher order gradients continuous or bounded is also discussed.

Let us consider a smooth (i.e. C∞ or Ck) nonlinear control system $$ \dot{x} = f(x,u)\quad x \in X, u \in U $$ (1) where f is a smooth mapping. For simplicity of exposition we will take X to be ℝn or an open subset of ℝn. (X could be an arbitrary manifold.) Furthermore we will make the (restrictive) assumption that U equals ℝm or an open subset of ℝm (or an arbitrary manifold without boundary). Let now L : X × U → ℝ and K : X → ℝ be smooth functions.

Helly and Klee type intersection theorems for finitary connected paved spaces

In a Hilbert space, we can introduce the notion of orthogonal coordinates through an orthogonal base, and these coordinates are the values of bounded linear functionals defined by the vectors of the base. This suggests that we consider continuous linear functionals, in a linear topological space, as generalized coordinates of the space. To ensure the existence of non-trivial continuous linear functionals in a general locally convex linear topological space, we must rely upon the Hahn-Banach extension theorems.

This paper is concerned with the existence and characterization of optimal growth paths in continuous time infinite horizon problems, and brings to bear on these issues mathematical techniques presented by the author in Chichilnisky (1977). These techniques, non-linear functional analysis in Hilbert (weighted L2 and Sobolev) spaces, permit generalization, clarification and simplification of existing results. In particular, the following points are of importance: (i) As Hilbert spaces are self-dual, continuous linear functionals on the commodity space are elements of the space, and thus have a totally natural interpretation as prices. We therefore obtain that commodity and price paths are in the same spaces, a useful property to prove existence in general equilibrium models. A failure to meet this condition has led to earlier work on characterization being troubled by the problem that supporting linear functionals are not always interpretable as prices, so that one cannot always give a price characterization of optimal and efficient paths. In the present framework this problem does not arise. (ii) A natural way to pose the problem of finding an optimal path is as one of maximizing a continuous function on a compact set. A major advantage of the techniques we use here is demonstrated by the fact that we are actually able to follow this approach. (iii) Because of (ii), it is possible to establish the existence of an optimal path without the use of any convexity assumptions on either the technology or the preferences. This is a major relaxation of earlier conditions, and enables existence results to be applied to models with increasing returns in production, non-convex preferences and endogenous population. For instance, in Lane (1977) both the feasible net production set and the preference ordering are in general non-convex due to the endogenous response of population growth to income. In Ryder and Heal (1973) and Wan (1970) non-convexities appear as well as in environmental and biological problems discussed for example in Clark (1976). In this latter case the logistic growth functions of biological populations introduces non-convexities in a natural way. (iv) It can be shown that the prices that arise within this framework provide a well-defined present value for every feasible program. This makes it possible to provide a full characterization of optimal and efficient paths in terms of profit maximization without any reference to transversality conditions. This characterization is thus analogous to that

It is proved that the density theorem of Arrow, Barankin, and Blackwell holds in a topological vector space equipped with a weakly closed convex cone to admit strictly positive continuous linear functionals. Moreover, several local versions of the Arrow, Barankin, and Blackwell theorem are given.

Consider a measurable Hilbert space (ℋ,𝓑) on which a measure μ is defined. Every continuous linear functional l(x) defined on ℋ is clearly 𝓑-measurable. It is known that if a sequence of continuous linear functionals l n (x) converges to a certain limit l(x) for all x then this limit will also be a continuous linear functional on ℋ. The situation, however becomes different if we require that l n (x) possess the limit not for all x but only on a set D such that μ(D)= 1. It is natural to refer to such limiting functions l(x) as 𝓑-measurable functionals. These functions, being limits of sequences of measurable functions will also be 𝓑-measurable. It follows from the relations $$ \mathop{{\lim }}\limits_{{n \to \infty }} {l_n}\left( {\alpha x + \beta y} \right) = \alpha \mathop{{\lim }}\limits_{{n \to \infty }} {l_n}(x) + \beta \mathop{{\lim }}\limits_{{n \to \infty }} {l_n}(y) $$ that the domain of definition D l of functional l(x) is a linear manifold and that l(x) is a linear (additive and homogeneous) functional. (We assume that the functional l(x) is defined wherever the corresponding limit exists.) Hereafter we shall consider non-degenerate measures μ such that μ(L) = 0 for any proper subspace L of the space ℋ.

A theory of discrete analysis