Basic sequences in the dual of a Fréchet space (Q2770431)

The main result of the paper under review (Theorem 2.4) establishes that, given a Fréchet space \(E\), the quotient map \(\varphi :E\rightarrow E/F\) lifts bounded sets (that is, for any \(B\subset E/F\) bounded there is \(A\subset E\) bounded with \( \varphi(A)=B\)) for every closed subspace \(F\) of \(E\) if and only if one of the following three conditions is satisfied: 1. \(E\) is a Banach space. 2. \(E\) is a Schwartz space. 3. \(E\) is the product of a Banach space with \(\omega\). NEWLINENEWLINENEWLINEThe problem of lifting of bounded sets for quotient maps between Fréchet spaces has interested many authors: A classical result of Palamodov, Merzon, and De Wilde [see \textit{M. De Wilde}, Bull. Soc. R. Sci. Liège 43, 299-301 (1974; Zbl 0292.46006)] says that a Fréchet space \(F\) is quasinormable if and only if for every Fréchet space \(E\) containing \(F\) as a topological subspace, the quotient map \(\varphi :E\rightarrow E/F\) lifts bounded sets. Conversely, by \textit{V. E. Cholodovskij} [Funkts. Anal. 7, 157-160 (1976; Zbl 0417.46002)], if \(E\) is a quasinormable Fréchet space and \(F\) is a closed subspace of \(E\) such that the quotient map \(\varphi :E\rightarrow E/F\) lifts bounded sets, then \(F\) is also quasinormable. \textit{J. Bonet} and \textit{S. Dierolf} [Proc. Edinb. Math. Soc., II. Ser. 36, No. 2, 277-281 (1993; Zbl 0492.46001)] proved that a quotient map between Fréchet spaces lifts bounded sets if and only if it satisfies the (weaker) assumption of lifting bounded sets with closure (equivalently, the strong transpose of the quotient map is a topological monomorphism). For more information about quasinormability and lifting of bounded sets, we refer the reader to Chapter 26 of the book of \textit{R. Meise} and \textit{D. Vogt} [``Introduction to functional analysis'', Clarendon Press, Oxford (1997; Zbl 0924.46002)]. NEWLINENEWLINENEWLINEIn the proof of Theorem 2.4 there is something missing since it refers to a non-existent Theorem 1.6. At this point, the proof should probably go as follows: If \(E\) is a Fréchet space with the property of bounded lifting then, by Theorem 2.2, \(E\) is quasinormable. The above mentioned result of Cholodovskij gives that every closed subspace \(F\) of \(E\) is quasinormable. The rest of the proof yields the desired result. NEWLINENEWLINENEWLINEThe main tool of Valdivia's arguments is a fine manipulation of weak\(^*\) basic sequences in duals of Fréchet spaces, which is essentially developed in the first section of the paper. This section contains results which are interesting on their own. NEWLINENEWLINENEWLINEThe final section is devoted to the study of totally reflexive Fréchet spaces. In particular, the author provides an easier proof of his result in \textit{M. Valdivia} [Math. Z. 200, 327-346 (1989; Zbl 0683.46008)] which says that a Fréchet space is totally reflexive if and only if it is isomorphic to a closed subspace of a countable product of reflexive Banach spaces.