Quojections and projective tensor products. (English) Zbl 0557.46003

Bonet, José

Arch. Math. 45, 169-173 (1985).

This article is devoted to the study of stability of barrelledness properties of projective tensor products of a Fréchet space E and a barrelled space F. The following results are proved:
(a) Let E be a Fréchet space. The following are equivalent:
1. Every quotient of E with a continuous norm is Banach space. (These Fréchet spaces are called quojections by Bellenot and Dubinsky).
2. E\({\hat \otimes}_{\pi}F\) is barrelled for every barrelled space F.
(b) Let E be a reflexive Fréchet space. The following are equivalent:
1. E is a quojection.
2. \(E\otimes_{\pi}F\) is barrelled for every barrelled space F which does not contain \(K^{(N)}\) complemented.
3. \(E{\hat \otimes}_{\pi}\lambda (A)'\!_ b\) is barrelled for every nuclear Köthe space \(\lambda\) (A) with a continuous norm.
Result (b) is obtained as a consequence of a characterization without the assumption of reflexivity.

MSC:

46A08	Barrelled spaces, bornological spaces
46M05	Tensor products in functional analysis
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.)
46A45	Sequence spaces (including Köthe sequence spaces)

Keywords:

stability of barrelledness properties of projective tensor products of a Fréchet space E and barrelled space F; quojections; nuclear Köthe space; continuous norm

References:

[1]	S. Bellenot andE. Dubinsky, Fréchet spaces with nuclear Köthe quotients. Trans. Amer. Math. Soc.273, 579-594 (1982). · Zbl 0494.46001
[2]	J. Bonet andP. Pérez Carreras, Some results on barrelledness in projective tensor products. Math. Z.185, 333-338 (1984). · Zbl 0538.46055 · doi:10.1007/BF01215044
[3]	A.Grothendieck, Produits tensoriels topologiques et espaces nuclearies. Mem. Amer. Math. Soc.16 (1955). · Zbl 0123.30301
[4]	H.Jarchow, Locally convex spaces. Stuttgart 1981. · Zbl 0466.46001
[5]	G.Köthe, Topological vector spaces I and II. New York-Heidelberg-Berlin 1969 and 1979.
[6]	V. B. Moscatelli, Fréchet spaces without continuous norms and without bases. Bull. London Math. Soc.12, 63-66 (1980). · doi:10.1112/blms/12.1.63
[7]	D.Vogt, Some results on continuous linear maps between Fréchet spaces. Funct. Anal., Surveys and Recent results III; K. D. Bierstedt and B. Fuchsteiner (Eds), Amsterdam 1984.
[8]	D. Vogt undM. J. Warner, Charakterisierung der Quotientenräume vons und eine Vermutung von Martineau. Studia Math.67, 225-240 (1980). · Zbl 0464.46010