Fréchet spaces with no infinite-dimensional Banach quotients. (English) Zbl 1261.46001
Whereas (closed) subspaces and (separated) quotients of reflexive Banach spaces are again reflexive, the situation is different for Fréchet spaces (\(=\) countable projective limits of Banach spaces): Grothendieck and Köthe constructed Fréchet-Montel spaces (where closed bounded convex sets are even compact and not only weakly compact as in the reflexive case) having non-reflexive Banach spaces as quotients. Grothendieck thus called a Fréchet space totally reflexive if all its quotients are reflexive, while M. Valdivia [Math. Z. 200, No. 3, 327–346 (1989; Zbl 0683.46008)] proved that these are precisely the countable projective limits of reflexive Banach spaces.
One of the main results of the present article is the construction of Fréchet-Montel spaces which are not totally reflexive but all of whose Banach quotients are finite-dimensional. The proof requires results of independent interest (also in the theory of Banach spaces) about block basic sequences in the dual of the James spaces \(J_p = \{ x\in c_0: \|x\|_{J_p} <\infty\}\), where \[ \|x\|_{J_p}^p= \frac{1}{2} \sup \{ |x_{k(n)}-x_{k(1)}|^p + \sum\limits_{i=1}^{n-1} |x_{k(i)}-x_{k(i+1)}|^p: n\geq 2,~1\leq k(1) < \cdots <k(n)\}. \] For example, the authors prove that every infinite-dimensional closed subspace of \(J_p'\) contains a subspace isomorphic to \(\ell_q\) (\(q\) being the conjugate exponent) which is complemented in \(J_p'\).
One of the main results of the present article is the construction of Fréchet-Montel spaces which are not totally reflexive but all of whose Banach quotients are finite-dimensional. The proof requires results of independent interest (also in the theory of Banach spaces) about block basic sequences in the dual of the James spaces \(J_p = \{ x\in c_0: \|x\|_{J_p} <\infty\}\), where \[ \|x\|_{J_p}^p= \frac{1}{2} \sup \{ |x_{k(n)}-x_{k(1)}|^p + \sum\limits_{i=1}^{n-1} |x_{k(i)}-x_{k(i+1)}|^p: n\geq 2,~1\leq k(1) < \cdots <k(n)\}. \] For example, the authors prove that every infinite-dimensional closed subspace of \(J_p'\) contains a subspace isomorphic to \(\ell_q\) (\(q\) being the conjugate exponent) which is complemented in \(J_p'\).
Reviewer: Jochen Wengenroth (Trier)
MSC:
| 46A25 | Reflexivity and semi-reflexivity |
| 46A04 | Locally convex Fréchet spaces and (DF)-spaces |
| 46B15 | Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces |
Citations:
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