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Generalizations of the theorems of Eberlein and Grothendieck on theprecompactness of subsets of function spaces are considered: if $X$ is acountably compact space and $C_p(X)$ is a space of continuous functions in thepointwise topology convergence, then any countably compact subspace of thespace $C_p(X)$ is precompact, that is, it has a compact closure. The paperprovides an overview of the results on this topic. It is proved that if apseudo-compact $X$ contains a dense Lindelof $\Sigma$-space, then pseudocompactsubspaces of the space $C_p(X)$ are precompact. If $X$ is the product Cechcomplete spaces, then bounded subsets of the space $C_p(X)$ are precompact.Results on the continuity of separately continuous functions were alsoobtained.

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