Geodesic
Sequential Compactness of X Implies a Completeness Property for C(X)
Canadian journal of mathematics, Tome 28 (1976) no. 1, pp. 207-210

Voir la notice de l'article provenant de la source Cambridge University Press

A locally convex Hausdorff topological vector space is said to be quasicomplete if closed bounded subsets of the space are complete, and von Neumann complete if closed totally bounded subsets are complete (equivalently, compact). Clearly quasi-completeness implies von Neumann completeness, and the converse holds in, for example, metrizable locally convex spaces. In this note we obtain a class of locally convex spaces for which the converse fails. Specifically, let X be a completely regular Hausdorff space, and let CC(X) denote the space of continuous real-valued functions on X, endowed with the compact-open topology. 
Rajagopalan, M.; Wheeler, R. F. Sequential Compactness of X Implies a Completeness Property for C(X). Canadian journal of mathematics, Tome 28 (1976) no. 1, pp. 207-210. doi: 10.4153/CJM-1976-026-9
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