Distinguished vector-valued continuous function spaces and injective tensor products

Abstract

The paper continues the study on the distinguished property of the space $C_{p}\left( X\right) $ (of real-valued continuous functions over a Tychonoff space $X$ in the pointwise topology) under the formation of tensor products towards the following research directions: $\left( i\right) $ the injective tensor product $C_{p}\left( X\right) \otimes _{\varepsilon }E$ of $C_{p}\left( X\right) $ and a real locally convex space $E$ (and its completion $C_{p}\left( X\right) \,\widehat{\otimes }_{\varepsilon }\,E$), and $\left( ii\right) $ the space $C_{p}\left( X,E\right) $ of all $E$-valued continuous functions (with a normed space $E$) endowed with the pointwise topology. This work leads also to a new characterization of distinguished Fréchet locally convex spaces $E$. We show, e.g., that if $C_{p}(X)$ is metrizable, then $E$ is distinguished if and only if metrizable $C_{p}\left( X\right) \otimes _{\varepsilon }E$ is distinguished.