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2019 Constructive Analysis on Banach spaces
Tepper L. Gill, Timothy Myers
Real Anal. Exchange 44(1): 1-36 (2019). DOI: 10.14321/realanalexch.44.1.0001

Abstract

Problems requiring analysis in higher-dimensional spaces have appeared naturally in electrical engineering, computer science, mathematics, physics, and statistics. In many cases, these problems focus on objects determined by an infinite number of parameters and/or are defined by functions of an infinite number of variables. They are currently studied using analytic, combinatorial, geometric and probabilistic methods from functional analysis. This paper is devoted to one of the important missing tools, a reasonable (or constructive) theory of Lebesgue measure for separable Banach spaces. A reasonable theory is one that provides: (1) a direct constructive extension of the finite-dimensional theory; and, (2) most (if not all) of the analytic tools available in finite dimensions. We approach this problem by embedding every separable Banach space into \(\mathbb{R}^\infty\) and use the unique \(\sigma {\text{-finite}}\) Lebesgue measure defined on this space as a bridge to the construction of a Lebesgue integral on every separable Banach space as a limit of finite-dimensional integrals. In our first application we define universal versions of Gaussian and Cauchy measure for every separable Banach space, which are absolutely continuous with respect to our Lebesgue measure. As our second application we constructively solve the diffusion equation in infinitely-many variables and introduce the interesting climate model problem of P. D. Thompson defined on infinite-dimensional phase space.

Citation

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Tepper L. Gill. Timothy Myers. "Constructive Analysis on Banach spaces." Real Anal. Exchange 44 (1) 1 - 36, 2019. https://doi.org/10.14321/realanalexch.44.1.0001

Information

Published: 2019
First available in Project Euclid: 27 June 2019

zbMATH: 07088960
MathSciNet: MR3951331
Digital Object Identifier: 10.14321/realanalexch.44.1.0001

Subjects:
Primary: ‎46G12
Secondary: 28C20

Rights: Copyright © 2019 Michigan State University Press

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Vol.44 • No. 1 • 2019
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