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 $\{a\in{\mathbb A}:\| x_{a}\| > \varepsilon\}$ 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, The complete separable extension property, J. Oper. Theory, 43 (2000), no. 2, 329-374]).