Real Analysis Exchange

Constructive Analysis on Banach spaces

Tepper L. Gill and Timothy Myers

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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.

Article information

Source
Real Anal. Exchange, Volume 44, Number 1 (2019), 1-36.

Dates
First available in Project Euclid: 27 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.rae/1561622429

Digital Object Identifier
doi:10.14321/realanalexch.44.1.0001

Mathematical Reviews number (MathSciNet)
MR3951331

Zentralblatt MATH identifier
07088960

Subjects
Primary: 46G12: Measures and integration on abstract linear spaces [See also 28C20, 46T12]
Secondary: 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11]

Keywords
measure on Banach space Lebesgue measure Cauchy measure Gaussian measure

Citation

Gill, Tepper L.; Myers, Timothy. Constructive Analysis on Banach spaces. Real Anal. Exchange 44 (2019), no. 1, 1--36. doi:10.14321/realanalexch.44.1.0001. https://projecteuclid.org/euclid.rae/1561622429


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