Banach Space Theory Research Articles

Contributions to the theory of the classical Banach spaces

The paper contains several results on the linear topological structure of the spaces C( K), K compact metric, and L p (0, 1), 1 ⩽ p < ∞. The topics which are studied include: complemented subspaces, special Schauder bases, and equivalent norms in these spaces.

Correction to the paper "On relatively disjoint families of measures, with some applications to Banach space theory" this volume pp. 13-36

Correction to the paper "On relatively disjoint families of measures, with some applications to Banach space theory" this volume pp. 13-36

Some aspects of the theory of Banach spaces

is defined. A normed space has a natural metric defined on it, viz., d(x, y) = 1) x y 1). If the metric space obtained in this way is complete, i.e., if every Cauchy sequence converges, the normed space is called a Banach space. The basic theory of Banach spaces forms at present an integral part of the graduate (and even advanced undergraduate) program in mathematics. It consists mainly of theorems like the Hahn-Banach theorem, the open mapping theorem, the uniform boundedness principle, and the Riesz theory of compact operators. These important theorems have the characteristics that they are useful for proving many theorems in classical analysis and that they hold for all Banach spaces. In the current investigations in Banach space theory an important role is played by the problem of classifying Banach spaces (i.e., finding invariants under certain equivalence relations). Of course, there is always

On relatively disjoint families of measures, with some applications to Banach space theory

On relatively disjoint families of measures, with some applications to Banach space theory

The structure of ideals and point derivations in Banach algebras of Lipschitz functions

Introduction. The main object of study in this paper is the space of all bounded complex-valued functions defined on a metric space (X, d) which satisfy a Lipschitz condition with respect to the metric d. With a suitable norm this collection of functions becomes a Banach algebra. This Banach algebra, which we shall call a Lipschitz algebra, will be denoted by Lip(X, d). It is the structure of Lip(X, d) with which we are primarily concerned here. Although the notion of Lipschitz continuity is very old and Lipschitz functions have been studied for many years, interest in the Banach space and Banach algebra theory of Lipschitz functions has not developed until quite recently. Very little is known about the Banach space properties of Lip (X, d). The following papers constitute all the work known to this writer which has been done in this area. Mirkil [9] and de Leeuw [4] have considered the space of periodic Lipschitz functions on the real line (or in other words, Lipschitz functions defined on the circle group). Mirkil was interested in the relation between the Lipschitz condition and the translation properties of functions, the fact that the functions were defined on a group being vital. de Leeuw was more directly concerned with the Banach space structure, his paper involving the study of certain dual spaces, extreme points, and isometries. In the long paper of Glaeser [6], a short chapter, somewhat off the main theme of his work, is devoted to the space of n-times continuously differentiable functions defined on a subset of Emn whose nth derivatives satisfy a Lipschitz condition. It is shown as a side result in the paper of Arens and Eells [1] that Lip(X,d) is always the dual space of some normed linear space. The only work known to us which treats Lip(X,d) as a Banach algebra was done by S. B. Myers [11]. His paper [11] is a summary of results, the proofs never published because of his untimely death in 1955. His interest in Lipschitz algebras was connected with the process of differentiation, and this interest we pursue in Chapter III of the present study. We supply proofs of many of the unproved statements of Myers, and in some cases extend his results to more general settings.

On the concept of categoricity

Mathematical theories may be divided into two classes. One includes theories built explicitly with a view to describing a certain definite mathematical domain, such as natural numbers, integers, real numbers, complex numbers, Euclidean plane, three-dimensional Euclidean space, etc. The other includes theories built for the purpose of grasping those properties that are common to many domains; this class covers, e. g., Boolean algebra, topology (closure algebra and its strength ened forms), lattice theory, group theory, ring theory, the theory of Banach spaces, set theory. The issue has naturally arisen with respect to the former class of theories, whether these theories really describe that domain they were intended to describe, or whether they describe, incidentally, a greater number of mathematical domains. An answer to this question was to be obtained from the examination of the cate goricity of these theories. Such examination used to be carried out in different ways, and hence the answers differed accordingly. First of all, the very concept of categoricity has, in the last thirty years, undergone considerable changes. The general intuition connected with that concept has remained the same: a set of sentences, Z, is categorical if it determines its model uniquely (up to isomorphism), that is if any two models for the set Z are, speaking freely, the same, or, speaking strictly, isomorphic. Yet at the time when that definition was first formulated (which seems to have taken place in 0. VEBLEN's paper of 1904) the concept of model still lacked precision; mathematical researches have subsequently shown that the said concept may be interpreted in various ways, and that for different interpretations of the concept of model the concept of categoricity acquires quite different meanings. At least four classes of models can be mentioned which may claim the right to be used in the definition of categoricity: (1) Models in the most general sense of the term. (2) Models of a definite power. (3) Models with an absolute interpretation of set-theoretical concepts. (4) Denumerable models of a special type (constructive or minimal). Each of these concepts of model and the related concept of categoricity will be discussed separately. Models of the type 3, when applied to the theories based

On The Closed Graph Theorem

The closed graph theorem is one of the deeper results in the theory of Banach spaces and one of the richest in its applications to functional analysis. This note contains an extension of the theorem to certain classes of topological vector spaces. For the most part, we use the terminology and notation of N. Bourbaki [1], contracting “locally convex topological vector space over the real or complex field” to “convex space”; here we confine ourselves to convex spaces.

Characterization of Analytic Functions in Banach Spaces

The concept of analyticity may be extended in various ways to functions from one complex Banach space to another. We may ask that the function be differentiable on one-dimensional (complex) subspaces; here one is led to the theory of the GAteaux differential. Or we may prescribe a seemingly much more powerful condition, namely, that the function possesses a development into (abstract) power series about each point of the domain of definition. Here the Fr6chet differential plays a decisive role. The characterization theorem which we are going to derive will serve to show that the functions which fall under the first definition but' not under the second are, from a certain point of view, to be considered as freaks, counter examples rather than examples. They are similar in character to, say, additive functions of a real variable which are not linear. For it turns out that only a very weak continuity property has to be added to the existence of the Gateaux differential in order to ensure the existence of the power series development called for by the second definition. This property-which we shall call B-continuity (and not, as we might, property of Baire)-aniounts to this: if a suitable set of the first category is dropped from the domain of definition, the remaining partial function shall be a continuous one. This property is preserved under passage to the limit and thus shared by large classes of functions. As a byproduct of the theory we obtain the solution of a problem which has long been unsolved. 0. We shall borrow freely from the theory of functions of a topological variable and occasionally from classical function theory. At a critical point we have to bring in the theory of abstract polynomials, and from the theory of Banach spaces and Banach space functions of one complex variable the following theorems will have to be used. (Unless there is explicit notice to the contrary we use only complex Banach spaces.) (0.1) Let F be a subset of a Banach space Y, and Y* the B-space of bounded linear functionals y* on Y to the complex numbers. If every functional y*(y) is bounded on F, the set F is bounded. (In the sense that the norms of its elements form a bounded set.) (0.2) Let f(v) be a function defined on an open set A of complex numbers, with values in a B-space Y.* If for every functional y* the numerical function y*[f(v)] is differentiable on A, the function f(v) is differentiable on A; its derivative is likewise differentiable. (0.3) If f(v) is differentiable for i v I < p, we have for these values of the argument the unique Taylor development f(v) = w [f (n)()]o/n!?. From the principle of the maximum awe need only the part: (0.4) If the function f(v) is differentiable for I v i _ p, then suplr.-p I f(0) I = supwrl=! If'v) 11 585