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We construct Zachary space in $\R^\infty$ and find that this is a Banachspace 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 $\R_I^\infty$ weconstruction $\mcB,$ where $\mcB $ is separable Banach space and finally weconstruct $\mcZ^p[\mcB]$.

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