Construction of the $ K{S^p}$ spaces on $\mathbb{R}^\infty$ and Separable Banach Spaces

Hemanta Kalita, Tepper L. Gill, Bipan Hazarika, Timothy Myers

Published 2020-02-21Version 1

The purpose of this paper is to construct a new class of separable Banach spaces $KS^p[\mathbb{R}^{\infty}], \; 1\leq p \leq \infty$. Each of these spaces contain the $ L^p[\mathbb{R}^\infty] $ spaces, as well as the space $\mfM[\mathbb{R}^\iy]$, of finitely additive measures as dense continuous compact embeddings. These spaces are of interest because they also contain the Henstock-Kurzweil integrable functions on $\mathbb{R}^\iy$. We will construct a canonical class of Kuelb-Steadman spaces $KS^p[\mathcal{B}], \; 1\leq p \leq \infty$, where $\mathcal{B}$ is separable Banach space. %and come again to $K{S^p}[\mathbb{R}^\infty].$ Finally, we offer a interesting approach to the Fourier transform on $K{S^p}[\mathbb{R}^{\infty}]$ and $KS^p[\mathcal{B}]$. % an application of $K{S^p}[\mathbb{R}^{\infty}].$ %Further we find $K{S^2}[\mathbb{R}^{\infty}] $ satisfies the requirement for Heisenberg and Schr\"{o}dinger representations.