(LF)-spaces, quasi-Baire spaces and the strongest locally convex topology
We relate the study of (LF)-spaces to several covering properties of locally convex spaces arising from the Baire category theorem. Explicitly, we consider the implications Baire \(\Rightarrow\) unordered Baire-like\(\Rightarrow (db)\Rightarrow Baire\)-like\(\Rightarrow quasi\)-Baire\(\Rightarrow barrelled.\) In the smallest ''variety'' of localy convex spaces, barrelled spaces are Baire-like. If \(\phi\) denotes a fixed \(\aleph_ 0\)-dimensional linear space endowed with the strongest locally convex topology, then, in the wider class of locally convex spaces not containing (an isomorphic copy of) \(\phi\), barrelled spaces are Baire-like. Here we show that in the still wider class of spaces not containing a complemented copy of \(\phi\), the notions of being quasi-Baire and barrelled coincide. Next, we exploit the class of (LF)-spaces to distinguish between these Baire-type properties. We partition all (LF)-spaces into three mutually disjoint, non-empty classes: \((LF)_ j\)-spaces \((i=1,2,3)\) and show that \((LF)_ 1,(LF)_ 2,(LF)_ 3\)-spaces are precisely the classes of (LF)- spaces which distinguish between barrelled and quasi-Baire, quasi-Baire and Baire-like, Baire-like and (db)-spaces, respectively. We provide nice characterizations of these three types of (LF)-spaces in terms of the space \(\phi\) and show that each of these distinguishing classes of (LF)- spaces is sufficiently rich. Elsewhere, we proved the abundant existence of \((LF)_ 3\)-spaces, which are precisely the class of metrizable (LF)- spaces, and we pointed out the intimate interaction between metrizable (LF)-spaces and the classical separable quotient problem. Specifically, we proved that every (LF)-space admits a separable, infinite-dimensional quotient; and a Fréchet space has a separable quotient if and only if it has a dense non-(db)-subspace. Here we prove several permanence properties for the various types of (LF)-spaces and the class of quasi- Baire spaces. While the quotient of an \((LF)_ i\)-space is an \((LF)_ j\)-space for some \(j\geq i\) or a Fréchet space, we show that every \((LF)_ 3\)-space admits a separable quotient which is an infinite- dimensional Fréchet space. The paper concludes with some remarks on completions of (LF)-spaces.
covering properties of locally convex spaces
scientific article; zbMATH DE number 3919060
scientific article