Minima and best approximations in constructive analysis

Hendtlass Hendtlass,Schuster Schuster

doi:10.4115/jla.2011.3.5

Abstract

Working in Bishop's constructive mathematics, we first show that min- ima can be defined as best approximations, in such a way as to preserve the compactness of the underlying metric space when the function is uniformly con- tinuous. Results about finding minima can therefore be carried over to the setting of finding best approximations. In particular, the implication from having at most one best approximation to having uniformly at most one best approximation is equivalent to Brouwer's fan theorem for decidable bars. We then show that for the particular case of finite-dimensional subspaces of normed spaces, these two notions do coincide. This gives us a better understanding of Bridges' proof that finite-dimensional subspaces with at most one best approximation do in fact have one. As a complement we briefly review how the case of best approximations to a convex subset of a uniformly convex normed space fits into the unique existence paradigm.

Highlights

The present paper is set in the realm of Bishop’s constructive mathematics (BISH)1
Even if X fails to be complete, the given data can be converted into an element of the completion of X : namely, into the Cauchy sequence in X. This observation has suggested a way [50]6 to get by in Bishop-style constructive mathematics without countable choice as proposed by Richman [44, 45], where completions are defined without sequences
Let X be a normed space, let x ∈ X, let E be a finite-dimensional subspace of X, and let S be a located convex subset of E

Summary

Introduction

The present paper is set in the realm of Bishop’s constructive mathematics (BISH). Bishop’s constructive framework is distinguished from classical—that is traditional— mathematics by a strict adherence to intuitionistic logic. A best approximation to z in S is nothing but a point of S at which the uniformly continuous function d ( · , z) : S → R attains its infimum dist (z, S). MIN Every uniformly continuous function on a compact metric space has a minimum, which would solve our problem with a single stroke of the pen, is a simple consequence of BWT. When does a uniformly continuous function on a compact metric space attain its infimum?. This observation has suggested a way [50]6 to get by in Bishop-style constructive mathematics without countable choice as proposed by Richman [44, 45], where completions are defined without sequences Being essentially folklore, this problem and its solution have some history. We conclude with a brief discussion of best approximations in a convex subset to a point in a uniformly convex normed space

Minima and best approximations

Finite-dimensional subspaces

Convex subsets