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Sequential completeness and regularity of inductive limits of webbed spaces. (English) Zbl 1075.46502

A. Grothendieck, in his 1955 Thèse [“Produits tensoriels topologiques et espaces nucléaires”, Ann. Inst. Fourier 4, 73–112 (1954; Zbl 0055.09705) and Mem. Am. Math. Soc. 16 (1955; Zbl 0064.35501) (Reprint 1963; Zbl 0123.30301); erratum Ann. Inst. Fourier 6, 117–120 (1955/56; Zbl 0064.35501)], proved that a separated inductive limit \(E= \text{ind}_n E_n\) of Fréchet spaces (called an (LF)-space) is locally complete (i.e., every bounded set in \(E\) is contained in a Banach disc) if and only if it is regular (i.e., every bounded set in \(E\) is contained and bounded in some \(E_n\)). It is an old question (attributed to Grothendieck and mentioned many times in the literature) whether regularity implies sequential completeness or completeness. In case the steps \(E_n\) are Banach spaces, Grothendieck proved that the inductive limit \(E\) is complete if and only if it is quasi-complete. Kučera, together with K. McKennon, has tried three times to give negative solutions (three counterexamples) to Grothendieck’s question [Int. J. Math. Math. Sci. 12, No. 3, 425–428 (1989; Zbl 0693.46063); ibid. 13, No. 4, 817–820 (1990; Zbl 0741.46001); ibid. 16, No. 4, 675–678 (1993; Zbl 0815.46005)]. As pointed out in the Zbl/MR-reviews, the first two articles contained essential mistakes, and the constructed inductive limits were not regular. In the reviewer’s opinion the third article mentioned above also contains a gap which was not mentioned in the review.
In Kučera’s previous article [Czech. Math. J. 51(126), No. 1, 181–183 (2001; Zbl 1079.46502)] and in the article under review, the authors venture on the search for a positive solution to Grothendieck’s problem. In the paper under review, they use strictly webbed spaces of De Wilde and completing strands. After a well-known lemma (better results can be seen in M. Valdivia [J. Lond. Math. Soc. (2) 35, 149–168 (1987; Zbl 0625.46006)]), the authors present a “proof” of a result which implies that every regular (LF)-space is sequentially complete. The main step of the proof (lines 13–20) in the proof on page 331 relies on an argument using duality and the Alaoglu theorem that would imply that every Fréchet space (hence every Banach space) is reflexive. This is well known to be false. Kučera’s proof in his article [op. cit.; Zbl 1079.46502)] relies exactly on the same argument.
In the humble opinion of the present reviewer, the problem of Grothendieck remains open. The reader interested in this problem should look at the important, recent work on regularity and completeness of (LF)-spaces by D. Vogt [Progress in Functional Analysis, North Holland Math. Stud. 170, 57–84 (1992; Zbl 0779.46005)] and J. Wengenroth [Studia Math. 120, No. 3, 247–258 (1996; Zbl 0863.46002)].

MSC:

46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
46A04 Locally convex Fréchet spaces and (DF)-spaces
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References:

[1] M. De Wilde: Closed Graph Theorem and Webbed Spaces. Pitman, 1978. · Zbl 0373.46007
[2] G. Köthe: Topological Vector Spaces II. Springer-Verlag, 1979. · Zbl 0417.46001
[3] W. Robertson: On the closed graph theorem with webs. Proc. London Math. Soc. 24 (1972), 692-738. · Zbl 0238.46005
[4] J. Kučera, C. Bosch: Bounded sets in fast complete inductive limits. Internat J. Math. 7 (1984), 615-617. · Zbl 0575.46001
[5] J. Kučera: Sequential completeness of \(LF\)-spaces. Czechoslovak Math. J 51 (2001), 181-183. · Zbl 1079.46502
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