Constructive Analysis in Infinitely Many Variables

Abstract

In this paper we investigate the foundations for analysis in infinitely-many (independent) variables. We give a topological approach to the construction of the regular $\sigma$-finite Kirtadze-Pantsulaia measure on $\mathbb{R}^\infty$ (the usual completion of the Yamasaki-Kharazishvili measure), which is an infinite dimensional version of the classical method of constructing Lebesgue measure on $\mathbb{R}^n$ (see [YA1], [KH] and [KP2]). First we show that von Neumann's theory of infinite tensor product Hilbert spaces already implies that a natural version of Lebesgue measure must exist on $\mathbb{R}^{\infty}$. Using this insight, we define the canonical version of $L^2[\mathbb{R}^{\infty}, \lambda_{\infty}]$, which allows us to construct Lebesgue measure on $\R^{\infty}$ and analogues of Lebesgue and Gaussian measure for every separable Banach space with a Schauder basis. When $\mathcal{H}$ is a Hilbert space and $\lambda_{\mathcal{H}}$ is Lebesgue measure restricted to $\mathcal{H}$, we define sums and products of unbounded operators and the Gaussian density for $L^2[\mathcal{H}, \lambda_{\mathcal{H}}]$. We show that the Fourier transform induces two different versions of the Pontryagin duality theory. An interesting new result is that the character group changes on infinite dimensional spaces when the Fourier transform is treated as an operator. Since our construction provides a complete $\s$-finite measure space, the abstract version of Fubini's theorem allows us to extend Young's inequality to every separable Banach space with a Schauder basis. We also give constructive examples of partial differential operators in infinitely many variables and briefly discuss the famous partial differential equation derived by Phillip Duncan Thompson [PDT], on infinite-dimensional phase space to represent an ensemble of randomly forced two-dimensional viscous flows.