Constructive analysis on Banach spaces. (English) Zbl 1429.46030

Gill, Tepper L.; Myers, Timothy

Real Anal. Exch. 44, No. 1, 1-36 (2019).

Summary: 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 [J. Fluid Mech. 55, 711–717 (1972; Zbl 0246.76053)] defined on infinite-dimensional phase space.

MSC:

46G12	Measures and integration on abstract linear spaces
28C20	Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
46N20	Applications of functional analysis to differential and integral equations

Keywords:

measure on Banach space; Lebesgue measure; Cauchy measure; Gaussian measure

Citations:

Zbl 0246.76053