### (DF)-Spaces of type and

#### Abstract

Some locally convex properties of the spaces 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 (e.g., see [S]).The following are our main results:
1. is a (DF)-space if and only if E is a (DF)-space.
2.For a (DF)-space 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 and if each is dominated by some , then (resp., ) is a (DF)-space if and only if E is a (DF)-space.
4. Let X be a locally compact and σ-compact space, and E a (DF)-space. Then is quasibarrelled if and only if (i) E is quasibarrelled and satisfies condition or (ii) the bounded subsets of E are metrizable and satisfies condition (D).

DOI Code:
10.1285/i15900932v10supn1p127

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