Weakly K-analytic spaces and the three-space property for analyticity

Let ( E , E ′ ) be a dual pair of vector spaces. The paper studies general conditions which allow to lift analyticity (or K-analyticity) from the weak topology σ ( E , E ′ ) to stronger ones in the frame of ( E , E ′ ) . First we show that the Mackey dual of a space C p ( X ) is analytic iff the space X is countable. This yields that for uncountable analytic spaces X the Mackey dual of C p ( X ) is weakly analytic but not analytic. We show that the Mackey dual E of an (LF)-space of a sequence of reflexive separable Fréchet spaces with the Heinrich density condition is analytic, i.e. E is a continuous image of the Polish space N N . This extends a result of Valdivia. We show also that weakly quasi-Suslin locally convex Baire spaces are metrizable and complete (this extends a result of De Wilde and Sunyach). We provide however a large class of weakly analytic but not analytic metrizable separable Baire topological vector spaces (not locally convex!). This will be used to prove that analyticity is not a three-space property but we show that a metrizable topological vector space E is analytic if E contains a complete locally convex analytic subspace F such that the quotient E / F is analytic. Several questions, remarks and examples are included.