Ferrando, J.C., Kąkol, J. (2022) "Distinguished vector-valued continuous function spaces and injective tensor products", Bulletin of the Belgian Mathematical Society – Simon Stevin, 28 (5):709–721

Juan Carlos Ferrando (Operations Research Center, University Miguel Hernández of Elche) and Jerzy Kąkol (Departamento de Física Teórica, Facultad de Ciencias Físicas, Universidad Complutense, Madrid)

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 $C_p\left(X\right){\otimes }_{\epsilon }E$ of $C_p\left(X\right)$ and a real locally convex space $E$ (and its completion $C_p\left(X\right){\hat{\otimes }}_{\epsilon }E$), 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 $C_p\left(X\right)$ is metrizable, then $E$ is distinguished if and only if metrizable $C_p\left(X\right){\otimes }_{\epsilon }E$ is distinguished.