Functional Analysis and Summability

The first four sections of this chapter form its core and include classical topics such as bounded linear transformations, the open mapping theorem, the closed graph theorem, the uniform boundedness principle, and the Hahn-Banach theorem. The chapter includes a good number of applications of the four fundamental theorems of functional analysis. Sections 6.5 and 6.6 provide a good account of the properties of the spectrum and adjoint operators on Banach spaces. They may be largely bypassed, since the treatment of the corresponding topics for operators on Hilbert spaces in chapter 7 is self-contained. The section on weak topologies is more advanced and may be omitted if a brief introduction is the goal. The chapter is enriched by such topics as the best polynomial approximation, the Hilbert cube, Gelfand’s theorem, Schauder bases, complemented subspaces, and the Banach-Alaoglu theorem.

The space $${\cal H}{\cal K}$$ of Henstock-Kurzweil integrable functions on [a, b] is the uncountable union of Frechet spaces $${\cal H}{\cal K}$$ (X). In this paper, on each Frechet space $${\cal H}{\cal K}$$ (X), an F-norm is defined for a continuous linear operator. Hence, many important results in functional analysis, like the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, hold for the $${\cal H}{\cal K}$$ (X) space. It is known that every control-convergent sequence in the $${\cal H}{\cal K}$$ space always belongs to a $${\cal H}{\cal K}$$ (X) space for some X. We illustrate how to apply results for Frechet spaces $${\cal H}{\cal K}$$ (X) to control-convergent sequences in the $${\cal H}{\cal K}$$ space. Examples of compact linear operators are given. Existence of solutions to linear and Hammerstein integral equations is proved.

The completeness of metric linear spaces has important consequences for linear maps between such spaces. These are obtained as deductions from Baire’s theorem. They divide themselves into two groups, namely, the open mapping theorem and the closed graph theorem on one side and the principle of uniform boundedness and the Banach—Steinhaus theorem on the other. We begin with the first of the above groups.

In this central chapter, the fundamental elements of functional analysis are presented. Although topological vector spaces are considered, essentially for the use of weak topologies, the focus is on normed spaces. Normed spaces in duality or metric duality form a convenient framework. The main pillars of functional analysis are presented: separation properties, the uniform boundedness theorem, the open mapping theorem, the closed graph theorem…Some special properties such as reflexivity, separability, and uniform convexity are given some attention. An account of spectral theory for linear operators is presented and the case of compact operators is considered. The chapter ends with a presentation of regulated functions and functions of bounded variation. This enables us to dispose of an elementary integration theory that is often sufficient for simple purposes.

Fuzzy bounded linear operators

Normed spaces are treated at length as well as techniques of Banach spaces for solving differential and integral equations. Classical results, such as the Open Mapping Theorem, the Closed Graph Theorem, the Hahn-Banach Theorem, and a few more, are given as well, establishing the core of the theory of bounded operators on Banach spaces. The chapter closes with a view towards generalizations of the theory in two directions, providing a glimpse of the theory of unbounded operators and of locally convex spaces.

This chapter develops the beginnings of abstract functional analysis, a subject designed to study properties of functions by treating the functions as the members of a space and formulating the properties as properties of the space. Section 1 defines Banach spaces as complete normed linear spaces and gives a number of examples of these. The space of bounded linear operators from one normed linear space to another is a normed linear space, and it is a Banach space if the range is a Banach space. Sections 2–3 concern Hilbert spaces. These are Banach spaces whose norms are induced by inner products. Section 2 shows that closed vector subspaces of such a space have orthogonal complements, and it shows the role of orthonormal bases for such a space. Section 3 concentrates on bounded linear operators from a Hilbert space to itself and constructs the adjoint of each such operator. Sections 4–6 prove the three main abstract theorems about the norm topology of general normed linear spaces—the Hahn–Banach Theorem, the Uniform Boundedness or Banach–Steinhaus Theorem, and the Interior Mapping Principle. A number of consequences of these theorems are given. The second and third of the theorems require some hypothesis of completeness. The topic of Hilbert and Banach spaces continues in Chapter IV of the companion volume, Advanced Real Analysis.

CHAPTER 23 - Banach Spaces and Topology

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.

On general open mapping theorems

Closed Graph Theorem and Open Mapping Theorem

We will establish order-preserving versions of the basic principles of functional analysis such as Hahn-Banach, Banach-Steinhaus, open mapping, and Banach-Alaoglu theorems.

A variation on the uniform boundedness theorem [2, p. 105] of functional analysis is obtained by replacing pointwise boundedness by a slightly stronger type of boundedness. This makes it possible to remove the category requirements from the set over which it is desired to have a collection of functions uniformly bounded. The concept of boundedness to be utilized is called boundedness on Cauchy sequences (see Definition 1). When a Cauchy sequence converges in the domain of a family of continuous functions, boundedness on a Cauchy sequence is equivalent to pointwise boundedness of the family at the limit point of the point of the sequence. When the domain of the continuous functions is a metric space, boundedness on the collection on all Cauchy sequences is equivalent to pointwise boundedness of the continuous extensions on the completion of the domain. The interesting case is when the domain is not metrizable because boundedness on Cauchy sequences does not necessarily imply pointwise boundedness of the extended functions on the completion or even on the sequential completion of the domain. The main result of the paper is given in Theorem 1 which is followed by six corollaries. The sixth corollary is especially interesting in stating that the closure of an infrabarrelled space in its second adjoint space is a barreled space. The theorem on uniform boundedness immediately suggests an absorption theorem which is Theorem 2 of the paper. Theorem 1 does not follow as a corollary of Theorem 2 as is the case of the classical uniform boundedness theorem [2, p. 105]. DEFINITION 1. A family H of continuous linear transformations with common domain a linear topological space E and ranges in a locally convex space F is bounded on a Cauchy sequence {Xn } in E if for every neighborhood V of the origin in F there exists a positive real number r with the property that for each f in H there is an Nf such that f (xn) is in r V for all n > Nf. The following theorem is stated in the context of the definition.

The Hahn–Banach theorem is another fundamental principle of functional analysis, which allows extending continuous linear functionals on a subspace while preserving continuity and linearity. An alternative version allows the separation of convex sets by hyperplanes. This chapter covers both versions together with their most important consequences.

Analysis of gene expression data using functional principal components