A strong barrelledness property for space C(X,E)


A locally convex space (lcs) E is called s-barrelled [DiK] if every sequentially closed linear map,i.e. with sequentially closed graph, of E into a Fréchet space,i.e. a metrizable and complete lcs, is continuous. Let E be a lcs, X a locally compact topological space, 𝛽 X its Stone-Cech compactification. If C(𝛽 X, E) is s-barrelled, then C(X, E) is s-barrelled iff X is realcompact, where all spaces of continuous functions are provided with the compact-open topology. Some remarks and corollaries are also included.

DOI Code: 10.1285/i15900932v14n2p199

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