Function bases for topological vector spaces

Abstract

Our main interest in this work is to characterize certain operator spaces acting on some important vector-valued function spaces such as $(V_{a}) _{c_{0}}^{a\in{\mathbb A}}$, by introducing a new kind basis notion for general Topological vector spaces. Where ${\mathbb A}$ is an infinite set, each $V_{a}$ is a Banach space and $( V_{a}) _{c_{0}}^{a\in{\mathbb A}}$ is the linear space of all functions $x\colon{\mathbb A} \rightarrow\bigcup V_{a}$ such that, for each $\varepsilon> 0$, the set $\{ a\in{\mathbb A}:\Vert x_{a}\Vert > \varepsilon\} $ is finite or empty. This is especially important for the vector-valued sequence spaces $( V_{i}) _{c_{0}}^{i\in{\mathbb N}}$ because of its fundamental place in the theory of the operator spaces (see, for example,[H. P. Rosenthal, {\it The complete separable extension property}, J. Oper. Theory, 43, No. 2, (2000), 329-374]).