Holomorphic germs on Tsirelson’s space

Mujica, Jorge; Valdivia, Manuel

doi:10.1090/S0002-9939-1995-1219730-5

Holomorphic germs on Tsirelson’s space
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by Jorge Mujica and Manuel Valdivia PDF

Proc. Amer. Math. Soc. 123 (1995), 1379-1384 Request permission

Abstract:

We show that if K is an arbitrary compact subset of the Banach space constructed by Tsirelson, then the space $\mathcal {H}(K)$ of all holomorphic germs on K, with its natural inductive limit topology, is totally reflexive.

References

Raymundo Alencar, On reflexivity and basis for $P(^mE)$, Proc. Roy. Irish Acad. Sect. A 85 (1985), no. 2, 131–138. MR 845536
Raymundo Alencar, Richard M. Aron, and Seán Dineen, A reflexive space of holomorphic functions in infinitely many variables, Proc. Amer. Math. Soc. 90 (1984), no. 3, 407–411. MR 728358, DOI 10.1090/S0002-9939-1984-0728358-5
R. M. Aron, C. Hervés, and M. Valdivia, Weakly continuous mappings on Banach spaces, J. Functional Analysis 52 (1983), no. 2, 189–204. MR 707203, DOI 10.1016/0022-1236(83)90081-2
R. M. Aron and J. B. Prolla, Polynomial approximation of differentiable functions on Banach spaces, J. Reine Angew. Math. 313 (1980), 195–216. MR 552473, DOI 10.1515/crll.1980.313.195
Richard M. Aron and M. Schottenloher, Compact holomorphic mappings on Banach spaces and the approximation property, J. Functional Analysis 21 (1976), no. 1, 7–30. MR 0402504, DOI 10.1016/0022-1236(76)90026-4
Klaus-Dieter Bierstedt and Reinhold Meise, Nuclearity and the Schwartz property in the theory of holomorphic functions on metrizable locally convex spaces, Infinite dimensional holomorphy and applications (Proc. Internat. Sympos., Univ. Estadual de Campinas, São Paulo, 1975) North-Holland Math. Studies, Vol. 12; Notas de Mat., No. 54, North-Holland, Amsterdam, 1977, pp. 93–129. MR 0632064
Klaus-Dieter Bierstedt and Reinhold Meise, Aspects of inductive limits in spaces of germs of holomorphic functions on locally convex spaces and applications to a study of $(H(U),\,\tau _{\omega })$, Advances in holomorphy (Proc. Sem. Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977) North-Holland Math. Stud., vol. 34, North-Holland, Amsterdam-New York, 1979, pp. 111–178. MR 520658

Preduals of the space of holomorphic functions on a Fréchet space

A. M. Davie, The approximation problem for Banach spaces, Bull. London Math. Soc. 5 (1973), 261–266. MR 338735, DOI 10.1112/blms/5.3.261
W. J. Davis, T. Figiel, W. B. Johnson, and A. Pełczyński, Factoring weakly compact operators, J. Functional Analysis 17 (1974), 311–327. MR 0355536, DOI 10.1016/0022-1236(74)90044-5
Seán Dineen, Complex analysis in locally convex spaces, Notas de Matemática [Mathematical Notes], vol. 83, North-Holland Publishing Co., Amsterdam-New York, 1981. MR 640093
Jeff D. Farmer, Polynomial reflexivity in Banach spaces, Israel J. Math. 87 (1994), no. 1-3, 257–273. MR 1286830, DOI 10.1007/BF02772998
Hikosaburo Komatsu, Projective and injective limits of weakly compact sequences of locally convex spaces, J. Math. Soc. Japan 19 (1967), 366–383. MR 217557, DOI 10.2969/jmsj/01930366
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056, DOI 10.1007/978-3-642-66557-8
Jorge Mujica, Spaces of germs of holomorphic functions, Studies in analysis, Adv. in Math. Suppl. Stud., vol. 4, Academic Press, New York-London, 1979, pp. 1–41. MR 546801
Jorge Mujica, Complex analysis in Banach spaces, North-Holland Mathematics Studies, vol. 120, North-Holland Publishing Co., Amsterdam, 1986. Holomorphic functions and domains of holomorphy in finite and infinite dimensions; Notas de Matemática [Mathematical Notes], 107. MR 842435
Jorge Mujica, Linearization of bounded holomorphic mappings on Banach spaces, Trans. Amer. Math. Soc. 324 (1991), no. 2, 867–887. MR 1000146, DOI 10.1090/S0002-9947-1991-1000146-2
Gilles Pisier, Counterexamples to a conjecture of Grothendieck, Acta Math. 151 (1983), no. 3-4, 181–208. MR 723009, DOI 10.1007/BF02393206

Applications of topological tensor products to infinite dimensional holomorphy

Not every Banach space contains an imbedding of

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M. Valdivia, A characterization of totally reflexive Fréchet spaces, Math. Z. 200 (1989), no. 3, 327–346. MR 978594, DOI 10.1007/BF01215650
George Willis, The compact approximation property does not imply the approximation property, Studia Math. 103 (1992), no. 1, 99–108. MR 1184105, DOI 10.4064/sm-103-1-99-108