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A characterization of classes of Fréchet spaces and the density condition. (English) Zbl 0814.46001

Fréchet spaces with St. Heinrich’s density condition were studied by K. D. Bierstedt and J. Bonet [Math. Nachr. 135, 149-180 (1988; Zbl 0688.46001)]; a Fréchet space satisfies the density condition iff the bounded sets of its strong dual are metrizable. Closed subspaces of Fréchet-Montel spaces and of Fréchet spaces isomorphic to a subspace of \(K^ J\times X\), \(X\) a Banach space, satisfy the density condition.
The authors prove the converse: Let \(E\) be a Fréchet space such that every closed subspace has the density condition. Then \(E\) is either Montel or \(E\cong K^ J\times\) Banach for some set \(J\).

MSC:

46A04 Locally convex Fréchet spaces and (DF)-spaces
46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)

Citations:

Zbl 0688.46001
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References:

[1] K. D. Bierstedt andJ. Bonet, Stefan Heinrich’s density condition for Fréchet spaces and the characterization of distinguished Köthe echelon spaces. Math. Nachr.135, 149–180 (1988). · Zbl 0688.46001 · doi:10.1002/mana.19881350115
[2] J.Bonet and J. C.Diaz, The density condition in subspaces and quotients of Fréchet spaces. Preprint.
[3] J.Diestel, Sequences and series in Banach spaces. Berlin-Heidelberg-New York 1984. · Zbl 0542.46007
[4] S. Heinrich, Ultrapowers of locally convex spaces and applications, I. Math. Nachr.118, 211–229 (1984). · Zbl 0601.46001
[5] S. Önal andT. Terzioğlu, Unbounded linear operators and nuclear Köthe quotients. Arch. Math.54, 576–581 (1990). · Zbl 0674.46001 · doi:10.1007/BF01188687
[6] S. Önal andT. Terzioğlu, A normability condition on locally convex space. Rev. Mat. Univ. Complut. Madrid4, 55–63 (1991).
[7] S.Önal and T.Terzioğlu, Concrete subspaces and quotient spaces of locally convex spaces and completing sequences. Dissertationes Math. (Rozprawy Mat.)318 (1992). · Zbl 0830.46002
[8] H. P. Rosenthal, A characterization of Banach spaces containing 1. Proc. Nat. Acad. Sci. USA71, 2411–2413 (1975). · Zbl 0297.46013 · doi:10.1073/pnas.71.6.2411
[9] T. Terzioğlu andM. Yurdakul, Restrictions of unbounded continuous linear operators on Fréchet spaces. Arch. Math.46, 547–550 (1986). · Zbl 0578.46002 · doi:10.1007/BF01195024
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