The density condition in subspaces and quotients of Fréchet spaces. (English) Zbl 0804.46002
The aim of this article is to investigate the density condition of subspaces and separated quotients of Fréchet spaces. This condition, introduced by S. Heinrich [Math. Nachr. 121, 211-229 (1985; Zbl 0601.46001)], was thoroughly studied by K. D. Bierstedt and J. Bonet [Math. Nachr. 135, 149-180 (1988; Zbl 0688.46001)]. They proved that a Fréchet space \(E\) has the density condition if and only if the bounded subsets of its strong dual are metrizable, and that a Köthe echelon space of order one, \(\lambda_ 1(I,A)\), is distinguished if and only if it has the density condition.
In this paper it is proved that every Fréchet space \(E\) which is neither Montel nor isomorphic to a subspace of \(X\times \omega\), with \(X\) a Banach and \(\omega= R^ \aleph\), must contain a closed subspace with basis and not satisfying the density condition, provided that \(E\) decomposes as \(F\oplus G\), where \(F\) and \(G\) are infinite-dimensional subspaces not isomorphic to \(\omega\). It is also proved that every Köthe echelon space of order \(p\) \((1< p<\infty)\) which is not quasinormable has a separated quotient with basis which does not satisfy the density condition.
In this paper it is proved that every Fréchet space \(E\) which is neither Montel nor isomorphic to a subspace of \(X\times \omega\), with \(X\) a Banach and \(\omega= R^ \aleph\), must contain a closed subspace with basis and not satisfying the density condition, provided that \(E\) decomposes as \(F\oplus G\), where \(F\) and \(G\) are infinite-dimensional subspaces not isomorphic to \(\omega\). It is also proved that every Köthe echelon space of order \(p\) \((1< p<\infty)\) which is not quasinormable has a separated quotient with basis which does not satisfy the density condition.
Reviewer: D.Zarnadze (Tbilisi)
MSC:
| 46A04 | Locally convex Fréchet spaces and (DF)-spaces |
| 46A45 | Sequence spaces (including Köthe sequence spaces) |
| 46A11 | Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.) |
Keywords:
density condition of subspaces and separated quotients of Fréchet spaces; Fréchet space; strong dual; Köthe echelon space; distinguished; quasinormableReferences:
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