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The separable quotient problem for barrelled spaces

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Book cover Functional Analysis and Related Topics, 1991

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1540))

Abstract

In this survey, we discuss the classical open problem:

“Does every infinite-dimensional Banach space admit a quotient (by a closed subspace) which is infinite-dimensional and separable?”

We furnish several equivalent formulations of this famous unsolved problem, in terms of certain Baire-type covering and barrel properties of locally convex spaces, and also in terms of the structure theory of (metrizable, normable) (LF)-spaces. We solve the corresponding “Separable Quotient Problem” for the class of (LF)-spaces in the affirmative, by actually constructing the separable quotient. Based on the Baire-type covering properties, all (LF)-spaces are partitioned into three mutually disjoint and sufficiently rich classes, called (LF)1, (LF)2 and (LF)3-spaces. These three classes are then characterized in terms of the sequence space ϕ. We show that every (LF)3-space admits a separable quotient, that is a Fréchet space. Every (LF)2 and (LF)3 space (more generally, all non-strict (LF)-spaces) possesses a defining sequence, each of whose members has a separable quotient. The intimate interaction between the Separable Quotient Problem for Banach spaces, and the existence of metrizable, as well as normable (LF)-spaces will be studied, resulting in a rich supply of metrizable, as well as normable (LF)-spaces. Finally, we discuss “Properly Separable” quotients in the setting of barrelled spaces.

Supported by NSERC Grant No. A-8772. Financial support from the organizers of the International Conference on Functional Analysis and Related Topics in memory of Professor Kôsaku Yosida, is gratefully acknowledged.

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Authors and Affiliations

  1. Department of Mathematics, Memorial University of Newfoundland, A1C 5S7, St. John's, Newfoundland, Canada

    P. P. Narayanaswami

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  1. P. P. Narayanaswami
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Hikosaburo Komatsu

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© 1993 Springer-Verlag

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Narayanaswami, P.P. (1993). The separable quotient problem for barrelled spaces. In: Komatsu, H. (eds) Functional Analysis and Related Topics, 1991. Lecture Notes in Mathematics, vol 1540. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085487

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  • DOI: https://doi.org/10.1007/BFb0085487

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  • Print ISBN: 978-3-540-56471-3

  • Online ISBN: 978-3-540-47565-1

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