On complemented copies of the space $c_0$ in spaces $C_p(X,E)$

TL;DR

This paper investigates when the space of continuous E-valued functions on a Tychonoff space X contains a complemented copy of c_0, extending classical results to broader classes of spaces and function spaces.

Contribution

It extends classical theorems by characterizing when $C_p(X,E)$ contains complemented copies of $c_0$, for a wide class of spaces, generalizing previous Banach and Fréchet space results.

Findings

Provides positive conditions for the existence of complemented c_0 in $C_p(X,E)$.

Shows the equivalence of complemented c_0 presence in $C_p(X,C_p(Y))$ and in $C_p(X)$, $C_p(Y)$ separately.

Extends classical theorems to broader classes of locally convex and Tychonoff spaces.

Abstract

We study the question for which Tychonoff spaces $X$ and locally convex spaces $E$ the space $C_p(X,E)$ of continuous $E$-valued functions on $X$ contains a complemented copy of the space $(c_0)_p=\{x\in\mathbb{R}^\omega\colon x(n)\to0\}$, both endowed with the pointwise topology. We provide a positive answer for a vast class of spaces, extending classical theorems of Cembranos, Freniche, and Doma\'nski and Drewnowski, proved for the case of Banach and Fr\'echet spaces $C_k(X,E)$. Also, for given infinite Tychonoff spaces $X$ and $Y$, we show that $C_p(X,C_p(Y))$ contains a complemented copy of $(c_0)_p$ if and only if any of the spaces $C_p(X)$ and $C_p(Y)$ contains such a subspace.