Skip to main content
main-content

Über dieses Buch

Many important reference works in Banach space theory have appeared since Banach's "Théorie des Opérations Linéaires", the impetus for the development of much of the modern theory in this field. While these works are classical starting points for the graduate student wishing to do research in Banach space theory, they can be formidable reading for the student who has just completed a course in measure theory and integration that introduces the L_p spaces and would like to know more about Banach spaces in general. The purpose of this book is to bridge this gap and provide an introduction to the basic theory of Banach spaces and functional analysis. It prepares students for further study of both the classical works and current research. It is accessible to students who understand the basic properties of L_p spaces but have not had a course in functional analysis. The book is sprinkled liberally with examples, historical notes, and references to original sources. Over 450 exercises provide supplementary examples and counterexamples and give students practice in the use of the results developed in the text.

Inhaltsverzeichnis

Frontmatter

1. Basic Concepts

Abstract
This chapter contains the basic definitions and initial results needed for a study of Banach spaces. In particular, the material presented in the first twelve sections of this chapter, with the exception of that of Section 1.5, is used extensively throughout the rest of this book. Section 1.13, though containing material that is very important in modern Banach space theory, is optional in the sense that the few results and exercises in the rest of the book that depend on this material are clearly marked as such.
Robert E. Megginson

2. The Weak and Weak Topologies

Abstract
The topology induced by a norm on a vector space is a very strong topology in the sense that it has many open sets. This has some advantages, especially since a function whose domain is such a space finds it particularly easy to be continuous, but it also has its disadvantages. For example, an infinite-dimensional normed space always has so many open sets that its closed unit ball cannot be compact. Because of this, many familiar facts about finite-dimensional normed spaces that are based on the Heine-Borel property cannot be immediately generalized to the infinite-dimensional case.
Robert E. Megginson

3. Linear Operators

Abstract
Linear operators have already received quite a bit of attention in this book, primarily as tools for probing the structure of normed spaces. The purpose of this chapter is to reverse that emphasis temporarily by studying linear operators between normed spaces as interesting objects in their own right, with the properties of normed spaces obtained in the first two chapters used as the tools for the study. Almost all of the attention will be on bounded linear operators, though there is also an interesting theory of unbounded ones; see, for example, Chapter 13 of [200] or Chapter VII, Section 9 of [67].
Robert E. Megginson

4. Schauder Bases

Abstract
Much of the theory of finite-dimensional normed spaces that has been presented in this book is ultimately based on Theorem 1.4.12, which says that every linear operator from a finite-dimensional normed space X into any normed space Y is bounded. A careful examination of the proof of that theorem shows that it essentially amounts to demonstrating that if x1,…, xn is a vector space basis for X, then each of the linear “coordinate functionals”α1x1 + … + α n x n ↦ α m , m = 1, …, n, is bounded; this can be seen from the nature of the norm /•/ used in the proof and the argument near the end of the proof that there is no sequence (z j ) in B X such that \(/{\text{I}}z_j / \geqslant j\) for each j. It should not be too surprising that many topological results about finite-dimensional normed spaces are ultimately based on the continuity of the members of the family \(\mathfrak{B}^\# \) of coordinate functionals for some basis \(\mathfrak{B}\) for the space, since it is an easy consequence of Proposition 2.4.8, Theorem 2.4.11, and the uniqueness of Hausdorff vector topologies for finite-dimensional vector spaces that the \(\mathfrak{B}^\# \) norm topology of the space is the topology of the space.
Robert E. Megginson

5. Rotundity and Smoothness

Abstract
When thinking of the closed unit ball of a normed space, it is tempting to visualize some round, smooth object like the closed unit ball of real Euclidean 2- or 3-space. However, closed unit balls are sometimes not so nicely shaped. Consider, for example, the closed unit balls of real ℓ 1 2 and ℓ 2 . Neither is round by any of the usual meanings of that word, since their boundaries, which is to say the unit spheres of the spaces, are each composed of four straight line segments. Also, neither is smooth along its entire boundary, since each has four corners. These features of the closed unit balls have a number of interesting consequences that cause the norms of these two spaces to behave a bit unlike that of real Euclidean 2-space. For example, if z1 and z2 are different points on any one of the four sides of one of these balls, then \(1\left\| {z_1 } \right\| = \left\| {z_2 } \right\| = \frac{1}{2}\left\| {z_1 } \right\| + \frac{1}{2}\left\| {z_2 } \right\| = \left\| {\frac{1}{2}z_1 + \frac{1}{2}z_2 } \right\|\) so equality is attained in the inequality \(\left\| {z_{1 + } z_2 } \right\| \leqslant \left\| {z_1 } \right\| + \left\| {z_2 } \right\|\) despite the fact that neither z1 nor z2 is a nonnegative real multiple of the other. Furthermore, the presence of the corners leads to the existence of multiple norming functionals for some points z of the unit sphere of each of these spaces, that is, norm-one members z* of the dual space such that \(z^* z = \left\| z \right\|\). To see why this would be, let Z be either of these two spaces and let z0 be one of the four corners of B z . Then there are infinitely many different straight lines that pass through z0 without intersecting the interior of the closed unit ball; let l1 and l2 be two of them. By Mazur’s separation theorem, there are members z 1 * and z 2 * of Z*, necessarily different, such that if j ∈ {1, 2}, then z j * z = 1 when zl j and z j * z ≤ 1 when zB z . It follows readily that z 1 * and z 1 * are both norming functionals for z0. As will be seen, it is precisely the presence of the corners or sharp bends in the unit sphere that caused this multiplicity of norming functionals for elements at the locations of the bends.
Robert E. Megginson

Backmatter

Weitere Informationen