(DF)-Spaces of type CB(X, E) and C\overline{V}(X, E)


Some locally convex properties of the spaces CB( X, E) of the bounded continuous functions on a completely regular Hausdorff space X with values in a (DF-space) E are studied and applied to the (DF)-spaces of type C\bar{V}(X,E) (e.g., see [S]).The following are our main results: 1.CB(X,E) is a (DF)-space if and only if E is a (DF)-space. 2.For a (DF)-space E, CB(X,E) is quasi barrelled if and only if either (i)X is pseudocompact and E is quasibarrelled or (ii) X is not pseudocompact and the bounded subsets of E are metrizaable. 3. If \mathcal V ⊂ C(X) and if each \bar{v}∈\bar{V} is dominated by some \tilde{v}∈ \bar{V}∩ C(X), then C\bar{V}(X,E) (resp., C\bar{V}(X)⨂_\varepsilon E) is a (DF)-space if and only if E is a (DF)-space. 4. Let X be a locally compact and σ-compact space, \mathcal V ⊂ C(X) and E a (DF)-space. Then C\bar{V}(X,E) is quasibarrelled if and only if (i) E is quasibarrelled and \mathcal V satisfies condition ( M, K) or (ii) the bounded subsets of E are metrizable and \mathcal V satisfies condition (D).

DOI Code: 10.1285/i15900932v10supn1p127

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