The Kakutani's precompactness lemma
Let \(X\) be a topological space and \(Z\) a pseudometric space. A mapping \(f: X \rightarrow Z\) is said to be K-continuous if for each \(\varepsilon > 0\) one can write \(X = U_1 \cup... \cup U_m\), \(m \in \mathbb{N}\), where each \(U_j\) is open in \(X\) and \(d_Z(f(x),f(y)) < \varepsilon\) for \(x\), \(y \in U_j\), \(j=1,\dots,m\). The notion of K-equicontinuity is defined in a similar way. A map \(f\) is K-continuous if and only if it is continuous and \(f(X)\) is precompact in \(Z\). The author shows the following extension of a precompactness lemma due to \textit{S.\ Kakutani} [J.\ Math.\ Soc.\ Japan 3, 228--231 (1951; Zbl 0043.32901)]: Let \(X\) and \(Y\) be topological spaces, \(Z\) a pseudometric space and \(f: X \times Y \rightarrow Z\) be a mapping such that \(f(X \times \{y\})\) is precompact for each \(y \in Y\) and \(f(\{x\} \times Y)\) is precompact for each \(x \in X\). Then, if all partial maps \(f_y: X \rightarrow Z\), \(f_y(x)=f(x,y)\) for \(y \in Y\), and \(f_x: Y \rightarrow Z\), \(f_x(y)=f(x,y)\) for \(x \in X\), are continuous, \(f: X \times Y \rightarrow Z\) is K-continuous. As a corollary, it follows that every separately continuous mapping \(f: X \times Y \rightarrow Z\) from a product \(X \times Y\) of two topological spaces which is a \(k\)-space into a pseudometric space \(Z\) is continuous. The result is then applied to prove an Arzelà-Ascoli theorem for continuous maps with precompact range, to derive a slight improvement of a precompactness theorem of \textit{A. Defant} and \textit{K. Floret} [Functional Analysis, Holomorphy and Approximation Theory, North-Holland Math. Stud. 86, 39--55 (1984; Zbl 0545.46007)] for families of linear operators, and a precompactness theorem for composition operators on the space of all continuous \(m\)-homogeneous polynomials or on the space of all entire mappings between complex locally convex spaces. In the last case, a variant of a result of \textit{R. M. Aron} and \textit{M. Schottenloher} [J. Funct.\ Anal.\ 21, 7--30 (1976; Zbl 0328.46046)] is obtained.
scientific article; zbMATH DE number 2107200
scientific article