Zachary spaces $\mathcal{Z}^p[\mathbb{R}^{\infty }]$ and separable Banach spaces

TL;DR

This paper constructs Zachary spaces in infinite-dimensional real spaces, demonstrating their Banach space properties, and explores their embeddings and applications to separable Banach spaces and functions of bounded mean oscillation.

Contribution

It introduces Zachary spaces in infinite dimensions, establishes their Banach space structure, and constructs related separable Banach spaces and embeddings.

Findings

Zachary space in $ ^ty$ is a Banach space of BMO functions.

BMO space is densely embedded in Zachary space.

Constructs a separable Banach space $$ and related Zachary space $Z^p[]$.

Abstract

We construct Zachary space in $\R^\infty$ and find that this is a Banach space of functions of bounded mean oscillation with order $p, 1\leq p \leq \infty$ containing the function of bounded mean oscillation $BMO[\R_I^\infty]$ as a dense continuous embedding. As an application of $\R_I^\infty$ we construction $\mcB,$ where $\mcB $ is separable Banach space and finally we construct $\mcZ^p[\mcB]$.