A characterization of totally reflexive Fréchet spaces. (English) Zbl 0683.46008
Following Grothendieck, a Fréchet space E is said to be totally reflexive if every separated quotient of E is reflexive. The author proves the beautiful theorem that a Fréchet space E is totally reflexive if and only if E is the projective limit of a sequence of reflexive Banach spaces (or, equivalently, if E is isomorphic to a closed subspace of a countable product of reflexive Banach spaces). From this characterization it is easy to deduce that the product \(E_ 1\times E_ 2\) of two totally reflexive Fréchet spaces \(E_ 1\) and \(E_ 2\) must itself be totally reflexive, which solves a problem in Grothendieck’s article on (F)- and (DF)-spaces.
The proof of (the difficult direction of the) main theorem involves (the Davis-Figiel-Johnson-Pełczyński factorization theorem for weakly compact operators and) the construction of a Schauder basis of a certain type for a suitable quotient E/F of any separable Fréchet space E which admits a decreasing fundamental sequence \((V_ n)_ n\) of closed absolutely convex 0-neighborhoods such that \(V^ 0_ 1\) is not weakly compact in \(E'_{V^ 0_ m}\) for arbitrary \(m\in {\mathbb{N}}.\)
This construction also shows that for each separable Fréchet space E which is not Schwartz, there is a closed subspace F such that E/F has a bounded Schauder basis.
The proof of (the difficult direction of the) main theorem involves (the Davis-Figiel-Johnson-Pełczyński factorization theorem for weakly compact operators and) the construction of a Schauder basis of a certain type for a suitable quotient E/F of any separable Fréchet space E which admits a decreasing fundamental sequence \((V_ n)_ n\) of closed absolutely convex 0-neighborhoods such that \(V^ 0_ 1\) is not weakly compact in \(E'_{V^ 0_ m}\) for arbitrary \(m\in {\mathbb{N}}.\)
This construction also shows that for each separable Fréchet space E which is not Schwartz, there is a closed subspace F such that E/F has a bounded Schauder basis.
Reviewer: K.D.Bierstedt
MSC:
46A25 | Reflexivity and semi-reflexivity |
46A04 | Locally convex Fréchet spaces and (DF)-spaces |
46A13 | Spaces defined by inductive or projective limits (LB, LF, etc.) |
46A11 | Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.) |
46A20 | Duality theory for topological vector spaces |
46M40 | Inductive and projective limits in functional analysis |
46B15 | Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces |
46A35 | Summability and bases in topological vector spaces |
Keywords:
problem of Grothendieck; weakly compact projective limits; quotients with a Schauder basis; Schauder basis with property P; separated quotient; projective limit of a sequence of reflexive Banach spaces; totally reflexive Fréchet spaces; Davis-Figiel-Johnson-Pełczyński factorization theorem for weakly compact operators; construction of a Schauder basis of a certain type for a suitable quotientReferences:
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