  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 $\mcZ^p[\mcB]$.
