### 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]).

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]).

#### Keywords

Biorthogonal systems; Schauder bases; generalization of bases; operators on function spaces; vector-valued function spaces; representation of operators

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