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

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 $(c_0{)}_p=\{x\in {\mathbb{R}}^{\omega }:x(n)\to 0\}$, both endowed with the pointwise topology. We provide a positive answer for a vast class of spaces, extending classical theorems of Cembranos, Freniche, and Domański and Drewnowski, proved for the case of Banach and Fréchet 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.

13 pages, fixed a number of typos, added some details