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Entire functions on \(c_ 0\). (English) Zbl 0538.46032

It is shown that every holomorphic function on the Banach space \(c_ 0\) of complex null sequences which is bounded on weakly compact subsets is bounded on bounded subsets of \(c_ 0\). This result answers a question of R. M. Aron, C. Hérves, M. Valdivia [J. Funct. Anal. 52, 189-204 (1983; Zbl 0517.46019)] in the negative. The result also shows that a holomorphic analogue of the following characterization of reflexive Banach spaces does not hold:
A Banach space E is reflexive if and only if every weakly continuous function on E is bounded on bounded subsets of E [cf. M. Valdivia, J. Funct. Anal. 24, 1-10 (1977; Zbl 0344.46004)].
Reviewer: M.Schottenloher

MSC:

46G20 Infinite-dimensional holomorphy
46A45 Sequence spaces (including Köthe sequence spaces)
32A15 Entire functions of several complex variables
46B10 Duality and reflexivity in normed linear and Banach spaces
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[1] Aron, R., Compact polynomials and compact differentiable mappings between Banach spaces, (Seminaire Pierre Lelong. Seminaire Pierre Lelong, 1974-1975. Seminaire Pierre Lelong. Seminaire Pierre Lelong, 1974-1975, Springer-Verlag Lecture Notes in Math. (1976)), 213-222, No. 524
[2] Aron, R.; Herves, C.; Valdivia, M., Weakly continuous mappings on Banach spaces, J. Funct. Anal., 52, 189-204 (1983) · Zbl 0517.46019
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