A note on quojections
\textit{S. F. Bellenot} and \textit{E. Dubinsky} [Trans. Am. Math. Soc. 273, 579-594 (1982; Zbl 0494.46001)] had proved that a separable Fréchet space F does not admit a nuclear Köthe quotient if and only if the space \(F^{'x}\) of all bounded linear forms on the strong dual \(F_ b'\) is a quojection (i.e. a projective limit of Banach spaces \((F_ n)_{n\in {\mathbb{N}}}\) where all the linking maps \(p_ n: F_ n\to F_{n-1}\) are surjective), and they had asked whether it is true that F is a quojection whenever \(F^{'x}\) is a quojection. In the present note this question is investigated. The authors first study a projective approach and prove I. Theorem. The following are equivalent i) There exists a Fréchet space F which is not a quojection such that \(F^{'x}\) is a quojection; ii) There exists a reduced projective sequence \(((F_ n)_{n\geq 1},(p_ n)_{n\geq 2})\) of Banach spaces such that all \(p_ n:\) \(F_ n\to F_{n-1}\) are injective but not surjective and such that for the bitranspose maps one has \(P''_{n+1}(F''_{n+1})+\ker p_ n''=F_ n''\) for all \(n\geq 2.\) Next they investigate an inductive approach and prove II. Theorem. The following are equivalent i) There exists a Fréchet space \(F\) which is not a quojection such that the strong dual \(F_ b'\) is a strict LB-space; ii) There exists a dual Banach space \(X'\) (with unit ball \(B\)) containing a strictly increasing sequence \((G_ n)_{n\in {\mathbb{N}}}\) of norm- closed, weakly dense linear subspaces \(G_ n\) such that the weak closure of \(B\cap G_ n\) is contained in \(G_{n+1}\) for all \(n\in {\mathbb{N}}.\) Two remarks should be added: 1) \textit{D. Vogt} proved 1986 in ''On two problems of Mityagin'' (to appear) that for a Fréchet space \(F\) the strong dual \(F_ b'\) is a strict LB-space if and only if \(F^{'x}\) is a quojection. 2) \textit{E. Behrends, P. Harmand} and the first author in ''On a problem of Bellenot and Dubinsky'' [Math. Ann. 275, 337-339 (1986; Zbl 0586.46001)] have constructed a dual Banach space \(X'\) satisfying condition ii) of the above II. theorem; in that way they obtained a separable Fréchet space \(F\) which admits no nuclear Köthe quotient and which at the same time is not a quojection.
scientific article; zbMATH DE number 4000712