Compact Resolutions and Analyticity

We consider the large class G of locally convex spaces that includes, among others, the classes of ( D F ) -spaces and ( L F ) -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 τ p of C ( X ) is analytic iff ( C ( X ) , ξ ) is separable and is covered by a compact resolution.