On complemented copies of the space c0 in spaces Cp(X,E)$C_p(X,E)$.

We study the question for which Tychonoff spaces X and locally convex spaces E the space Cp(X,E)$C_p(X,E)$ of continuous E‐valued functions on X contains a complemented copy of the space (c0)p={x∈Rω:x(n)→0}$(c_0)_p=\lbrace x\in \mathbb {R}^\omega : x(n)\rightarrow 0\rbrace$, 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 Ck(X,E)$C_k(X,E)$. Also, for given infinite Tychonoff spaces X and Y, we show that Cp(X,Cp(Y))$C_p(X,C_p(Y))$ contains a complemented copy of (c0)p$(c_0)_p$ if and only if any of the spaces Cp(X)$C_p(X)$ and Cp(Y)$C_p(Y)$ contains such a subspace. [ABSTRACT FROM AUTHOR]