Asplund spaces and the finest locally convex topology

Abstract: In our previous paper we systematized several known equivalent definitions of Fr\'echet (G^ ateaux) Differentiability Spaces and Asplund (weak Asplund) Spaces. As an application, we extended the classical Mazur's theorem, and also proved that the product of any family of Banach spaces $(E_{\alpha})$ is an Asplund lcs if and only if each $E_{\alpha}$ is Asplund. The actual work continues this line of research in the frame of locally convex spaces, including the classes of Fr\'echet spaces (i.e. metrizable and complete locally convex spaces) and projective limits, quojections, $(LB)$-spaces and $(LF)$-spaces, as well as, the class of free locally convex spaces $L(X)$ over Tychonoff spaces $X$. First we prove some "negative" results: We show that for every infinite Tychonoff space $X$ the space $L(X)$ is not even a G^ ateaux Differentiability Space (GDS in short) and contains no infinite-dimensional Baire vector subspaces. On the other hand, we show that all barrelled GDS spaces are quasi-Baire spaces, what implies that strict $(LF)$-spaces are not GDS. This fact refers, for example, to concrete important spaces $D^{m}(\Omega)$, $D(\Omega)$, $D(\mathbb{R}^{\omega})$. A special role of the space $\varphi$, i.e. an $\aleph_{0}$-dimensional vector space equipped with the finest locally convex topology, in this line of research has been distinguished and analysed. It seems that little is known about the Asplund property for Fr\'echet spaces. We show however that a quojection $E$, i.e. a Fr\'echet space which is a strict projective limit of the corresponding Banach spaces $E_n$, is an Asplund (weak Asplund) space if and only if each Banach space $E_n$ is Asplund (weak Asplund). In particular, every reflexive quojection is Asplund. Some applications and several illustrating examples are provided.