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

Abstract

AbstractWe study the question for which Tychonoff spaces X and locally convex spaces E the space of continuous E‐valued functions on X contains a complemented copy of the space , 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 . Also, for given infinite Tychonoff spaces X and Y, we show that contains a complemented copy of if and only if any of the spaces and contains such a subspace.