Grothendieck’s Theorem on the Precompactness of Subsets of Functional Spaces over Pseudocompact Spaces

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 on $$X$$ in the topology of pointwise 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 pseudocompact $$X$$ contains a dense Lindelöf $$\Sigma$$ -space, then pseudocompact subspaces of the space $$C_p(X)$$ are precompact. If $$X$$ is the product Čech complete spaces, then bounded subsets of the space $$C_p(X)$$ are precompact. Results on the continuity of separately continuous functions are also obtained.