Joe Diestel

Geometry of Banach spaces : selected topics

Support functionals for closed bounded convex subsets of a Banach space.- Convexity and differentiability of norms.- Uniformly convex and uniformly smooth Banach spaces.- The classical renorming theorems.- Weakly compactly generated banach spaces.- The Radon-Nikodym theorem for vector measures.

OBSERVATIONS ABOUT THE PROJECTIVE TENSOR PRODUCT OF BANACH SPACES, I—

Abstract A sequential description of the projective tensor product lp⊗X, is given. This description allows us to show the inclusion lp⊗X, into lp⊗X, is a semi-embedding. As a consequence, if X has the Radon-Nikodym property so does lp⊗X, A discussion of difficulties with Lp(0, 1) ⊗X, follows.

SOME OBSERVATIONS ABOUT THE SPACE OF WEAKLY CONTINUOUS FUNCTIONS FROM A COMPACT SPACE INTO A BANACH SPACE

For emergency cooling of a nuclear reactor plant, there is provided a primary flow circuit for cooling gas through the reactor core, and then through the primary sides of a reheater and a main steam generator, to a main circulator pump which returns the gas to the core. Another flow circuit is provided which conducts condensed water from a turbine condenser to the secondary side of a main steam generator, and conducts the steam produced therein through a high pressure part of a main turbine and then to the secondary side of the reheater and to the low pressure part of the main turbine and to the condenser. Condensed water from the condenser is also pumped into an auxiliary generator parallel with the main steam generator where it is vaporized and the steam produced is conducted to the secondary part of the reheater 9, and then to a turbine which drives the main circulator. A shunt conduit in connection with a shunt valve is arranged to connect the outlet of the secondary side of the main steam generator to the inlet of the drive turbine of the main circulator.

Chapter 11 – Operator Ideals

This chapter explains the theory of operator ideals that was born in 1941 with the observation of Calkin. Schatten and von Neumann actually dealt with more general classes of operators; the so-called cross-spaces of linear transformations. They phrased their results in the language of direct (now tensor) products of the Banach spaces. To stress the analogy with operator ideals, Gothic bold lowercase letters will be used to denote sequence ideals. The local theory of Banach spaces deals with those properties that can be expressed in terms of finite dimensional spaces. The most usual way is to consider the quantitative relations that are assumed to hold uniformly for any choice of n elements and/or functionals. Grothendiecks theorem has many applications in Banach space geometry. Determining precisely when a concrete operator of prescribed form belongs to a given ideal often provides considerable enlightenment about the operator, its domain and codomain, and the ideal.

The Joys of Haar Measure

From the earliest days of measure theory, invariant measures have held the interests of geometers and analysts alike, with the Haar measure playing an especially delightful role. The aim of this book is to present invariant measures on topological groups, progressing from special cases to the more general. Presenting existence proofs in special cases, such as compact metrizable groups, highlights how the added assumptions give insight into just what the Haar measure is like; tools from different aspects of analysis and/or combinatorics demonstrate the diverse views afforded the subject. After presenting the compact case, applications indicate how these tools can find use. The generalisation to locally compact groups is then presented and applied to show relations between metric and measure theoretic invariance. Steinlages approach to the general problem of homogeneous action in the locally compact setting shows how Banachs approach and that of Cartan and Weil can be unified with good effect. Finally, the situation of a nonlocally compact Polish group is discussed briefly with the surprisingly unsettling consequences indicated. The book is accessible to graduate and advanced undergraduate students who have been exposed to a basic course in real variables, although the authors do review the development of the Lebesgue measure. It will be a stimulating reference for students and professors who use the Haar measure in their studies and research.

GROTHENDIECK SPACES AND VECTOR MEASURES

Publisher Summary This chapter explains certain, at first unrelated, phenomena in the theory of finitely additive measures. Bounded additive maps defined on a sigma–algebra with values in either a separable Banach space or a Banach space not containing c0 are almost countably additive, that is, they add up every sequence of disjoint sets—perhaps not to the union. These efforts have led to the consideration of a special class of Banach spaces, the so-called Grothendieck spaces. The relationship of Grothendieck spaces to vector measures is simple: every vector measure lies on a suitably chosen Grothendieck space. This relationship was exploited to obtain results in vector measures. The chapter focuses on Grothendieck spaces and presents a theorem that gives several new characterizations of this class of spaces. This theorem is then applied to yield some results in vector measures.

CONSTRAINED CONTROLLABILITY IN NON REFLEXIVE BANACH SPACES

Abstract Throughout the literature on optimal control in Banach spaces, hypotheses like “separable and reflexive” are frequently encountered. In this note we consider one such case, studied by Peichl and Schappacher. Using techniques from Banach space theory and the theory of vector measures, we show how to remove the hypothesis of reflexivity and translate the problem of controls to one about the strong continuity of an adjoint semigroup on the positive real axis.

CHAPTER 9 – The Riesz Theorem

This chapter discusses about the Riesz theorem. The use of the Riesz theorem that is used in tandem with the Hahn–Banach theorem is examined in the chapter. One of the first applications of the Riesz theorem is in characterizing weakly convergent sequences in metric spaces. The partnership of the Riesz Theorem with various forms of the Hahn–Banach theorem is the most powerful in all abstract analysis. The most basic form of the Hahn–Banach theorem extending bounded linear functional from a subspace of a Banach space to the whole space without changing the norm is analyzed in the chapter. The Banachs version of the extension theorem is applied wherein a linear functional that is dominated by a positively homogeneous, subadditive functional is extended to the whole space with domination remaining. One remarkable consequence of Choquets theorem is the characterization of weakly null sequences in Banach spaces. It is shown that between Hilbert spaces, the 2-summing operators and Hilbert–Schmidt operators are precisely the same, while the trace-class coincides with the nuclear operators.

The Classical Banach Spaces

To this juncture, we have dealt with general theorems concerning the nature of sequential convergence and convergence of series in Banach spaces. Many of the results treated thus far were first derived in special cases, then understood to hold more generally. Not too surprisingly, along the path to general results many important theorems, special in their domain of applicability, were encountered. In this chapter, we present more than a few such results.

The Eberlein-Šmulian Theorem

We saw in the previous chapter that regardless of the normed linear space X, weak* closed, bounded sets in X* are weak* compact. How does a subset K of a Banach space X get to be weakly compact? The two are related. Before investigating their relationship, we look at a couple of necessary ingredients for weak compactness and take a close look at two illustrative nonweakly compact sets.