Grothendieck's theorem on the precompactness of subsets functional spaces over pseudocompact spaces

Abstract

Generalizations of the theorems of Eberlein and Grothendieck on the precompactness of subsets of function spaces are considered: if $X$ is a countably compact space and $C_p(X)$ is a space of continuous functions in the pointwise topology convergence, then any countably compact subspace of the space $C_p(X)$ is precompact, that is, it has a compact closure. The paper provides an overview of the results on this topic. It is proved that if a pseudo-compact $X$ contains a dense Lindelof $\Sigma$-space, then pseudocompact subspaces of the space $C_p(X)$ are precompact. If $X$ is the product Cech complete spaces, then bounded subsets of the space $C_p(X)$ are precompact. Results on the continuity of separately continuous functions were also obtained.