Compact Resolutions and Analyticity

Abstract

We consider the large class G of locally convex spaces that includes, among others, the classes of (DF)-spaces and (LF)-spaces. For a space E in class G we have characterized that a subspace Y of (E,σ(E,E′)), endowed with the induced topology, is analytic if and only if Y has a σ(E,E′)-compact resolution and is contained in a σ(E,E′)-separable subset of E. This result is applied to reprove a known important result (due to Cascales and Orihuela) about weak metrizability of weakly compact sets in spaces of class G. The mentioned characterization follows from the following analogous result: The space C(X) of continuous real-valued functions on a completely regular Hausdorff space X endowed with a topology ξ stronger or equal than the pointwise topology τp of C(X) is analytic iff (C(X),ξ) is separable and is covered by a compact resolution.

Keywords

• compact resolution
• analytic space
• locally convex space
• weak metrizability
• <i>C_p</i>(<i>X</i>)-spaces