Selected Papers of Errett Bishop

The following sections are included:

• CONTENTS

• VITA*

• INTRODUCTION

• BIBLIOGRAPHY OF THE PUBLICATIONS OF ERRETT BISHOP (LIST I)

The following sections are included:

• Introduction

• Vector-valued measures

• Measures associated with operators

• Application to scalar transformations

• Functional calculus

• Weak T-measures

• REFERENCES

The following sections are included:

• Introduction and preliminaries

• Preparatory lemmas

• The main theorem and its consequences

• REFERENCES

The structure of a (finite, complex-valued, Borel) measure on the unit circle |w|= 1 which is orthogonal to all polynomials in w is well known. It is given by the following theorem,¹ which is partially due to F. and M. Riesz, and partially due to Fatou…

The following sections are included:

• Introduction and preliminaries

• The representation of μ by g(z) dz

• The structure of μ

• Applications

• REFERENCES

The work on the theory of approximations initiated by Weierstrass and continued by Walsh, Keldysh, and Lavrentiev, among others, has culminated in the following theorem of Mergelyan (See Mergelyan [3]): Given any compact subset C of the complex plane, which does not separate the plane, and given any continuous function f on C which is analytic interior to C, then f can be approximated uniformly on C by polynomials…

The following sections are included:

• Introduction

• Definitions and preliminaries

• Duality theory of type 4

• The other duality theories

• REFERENCES

The following sections are included:

• Introduction

• The minimal boundary

• Representation by measures

• Applications

• REFERENCES

In what follows, C will be a compact Hausdorff space, and will be a uniformly closed separating algebra of continuous complexvalued functions on C. That is, sums, products, and complex multiples of elements of are in , uniform limits of elements of are in , and for distinct points x and y in C there exists f in with f(x)≠f(y). Šilov (see [1]) has shown that there exists a smallest closed subset B of C, called the Šilov boundary of such that for each f in there exists x in B with |f(x)| = ∥f∥, where ∥f∥ = max {|f(y)| : y∈C}. The following theorem generalizes this result, in case C is metric…

The following sections are included:

• INTRODUCTION

• THE BASIC DEFINITIONS

• REPRESENTATION OF LINEAR FUNCTIONALS IN THE SEPARABLE CASE

• EXTREME POINTS AND THE CHOQUET THEOREM

• REPRESENTATION OF LINEAR FUNCTIONALS IN THE GENERAL CASE

• ALGEBRAS

• EXAMPLES

• BIBLIOGRAPHY

The following sections are included:

• Introduction

• Proof of the Theorem

• BIBLIOGRAPHY

The following sections are included:

• Introduction

• The classes M and H

• The main theorems

• Remarks

• REFERENCES

The following sections are included:

• Introduction

• Proof of the theorem

• REFERENCES

A real or complex normed space is subreflexive if those functionals which attain their supremum on the unit sphere S of E are normdense in E*, i.e., if for each f in E* and each ∈>0 there exist g in E* and x in S such that |g(x)| = ∥g∥ and ∥f−g∥<∈. There exist incomplete normed spaces which are not subreflexive [1]¹ as well as incomplete spaces which are subreflexive (e.g., a dense subspace of a Hilbert space). It is evident that every reflexive Banach space is subreflexive. The theorem mentioned in the title will be proved for real Banach spaces; the result for complex spaces follows from this by considering the spaces over the real field and using the known isometry between complex functionals and the real functionals defined by their real parts…

The following sections are included:

• Introduction

• Dimensions of level sets of mappings

• Analytic polyhedra and weak holomorphic convexity

• Special analytic polyhedra and integration

• Mappings into higher dimensional spaces

• REFERENCES

The following sections are included:

• Introduction

• Preliminaries

• Analytic sets in Stein manifolds

• The resolution of Cousin distributions

• REFERENCES

The following sections are included:

• Introduction

• Partially analytic spaces

• Special frames

• Contractions of analytic spaces

• Imbeddings and related maps

• REFERENCES

The following sections are included:

• Introduction

• Proof of Theorem

• REFERENCES

We wish to extend certain results in the theory of analytic functions of several complex variables to the case of analytic functions with values in a Frechet space F. To do this, we prove (Theorem 1 below) that such a function φ has an expansion of the form

where {P_n} is a sequence of continuous mutually annihilating projections on F whose ranges are all one-dimensional subspaces of F. This representation reduces the study of φ, for many purposes, to the study of the functions , which are essentially scalar-valued analytic functions. We actually prove the stronger (and more useful) result that if {φ_k} is a sequence of analytic functions with values in F then a single sequence {P_n} can be found to give an expansion (*) for every φ_k. Expansions of vector-valued functions of a different type have been considered by Grothendick [6]…

The following sections are included:

• Introduction

• Functions rational over

• Conditions for analyticity of the spectrum

• Proofs of Theorems 1 and 2

• REFERENCES

The following well-known separation theorem is basic to the considerations of this paper…

The following sections are included:

• Introduction

• Locally convex topological algebras

• Algebras of analytic functions

• Manifolds with global local coordinates

• The case of a Riemann surface

• REFERENCES

Recent years have seen much effort put into attempts to develop an “abstract” function theory. In the complex domain, this has led to the study of the structure of uniform algebras. A uniform algebra is a family of continuous complex-valued functions on a compact Hausdorff space X, which contains the function 1, which is closed with respect to the algebraic operations of addition and multiplication by complex scalars, which is topologically closed in the uniform norm, and which distinguishes points of X. Succinctly, is a closed separating unitary subalgebra of the Banach algebra C(X) of all continuous complex-valued functions on X. Standard examples are obtained by taking X to be a compact subset of complex Euclidean space Cⁿ and to be the closed subalgebra of C(X) generated by the constants and the coordinate functions on Cⁿ…

The following sections are included:

• INTRODUCTION

• CONVERGENCE OF ANALYTIC SETS

• CONDITIONS FOR REMOVABLE SINGULARITIES

• CAPACITY

• REFERENCES

Recently concepts from the theory of Banach algebras have been increasingly useful in applications to complex analysis. Here we give a quick survey of some, but by no means all, of these applications. The algebras which occur are algebras of complex valued functions and they are given the uniform norm with respect to some compact set K. Since the usual terms for these algebras (sup. norm algebras or function algebras) are not euphonious, we shall call them uniform algebras…

We consider a k-dimensional differentiable submanifold M^k of n-dimensional complex coordinate space Cⁿ, and interest ourselves in the relation of the submanifold M to the analytic subvarieties of Cⁿ…

The method of Banach algebras, like so many methods of functional analysis, gives with one hand and takes away with the other. It gives a powerful technique for existence proofs in certain parts of analysis, but it takes away the possibility of constructing the objects asserted to exist…

The theorem of Lebesgue that a function of bounded variation has a derivative almost everywhere, the ergodic theorem of Birkhoff, and the martingale theorem have the common property of being nonconstructive; they are not true in intuitionist mathematics. (See Brouwer [1] for an example of a function of bounded variation that does not have a derivative almost everywhere.) This paper developed from an attempt to save the phenomena…

In [1] an upcrossing inequality underlying Lebesgue’s theorem that a function of bounded variation has a derivative a.e. was derived. As corollaries, Doob’s upcrossing inequality for martingales and an upcrossing inequality underlying Birkhoff’s ergodic theorem were obtained. In [2], by entirely different methods, an upcrossing inequality underlying a generalization of the Chacon-Ornstein ergodic theorem was obtained, and upcrossing inequalities for Lebesgue’s theorem and the martingale theorem derived as corollaries. In this paper, using different methods from both [1] and [2], we derive a more general upcrossing inequality yet. This inequality implies both the Chacon-Ornstein and Dunford-Schwartz ergodic theorems. In fact, it implies a common generalization [3] of these theorems, due to Chacon, as a special case…

The constructive point of view is that all mathematics should have numerical meaning. In other words, every mathematical theorem should admit an ultimate interpretation to the effect that certain finite computations within the set of positive integers will give certain results. In contrast, classical (that is, contemporary) mathematics is idealistic: there is no requirement that theorems and their proofs have a numerical meaning, or any predictive content whatever. For instance, the theorem that the real numbers can be well-ordered is evocative (or idealistic), rather than descriptive (or constructive). So is the theorem that a bounded monotone sequence of real numbers converges…