Ferrera Cuesta, Juan and Prieto Yerro, M. Ángeles
(1994)
*The strict topology on spaces of bounded holomorphic functions.*
Bulletin of the Australian Mathematical Society, 49
(2).
pp. 249-256.
ISSN 0004-9727

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## Abstract

The authors consider the space of bounded holomorphic functions on the open unit ball of a Banach space with the natural analogue of the strict topology β for the one- (or finite-)dimensional case. This can be defined by means of weighted seminorms or as the mixed topology (in the sense of Wiweger) of the supremum norm and the topology of uniform convergence on subsets of the unit ball which are bounded away from its complement. The natural analogues of some elementary properties of the one-dimensional case are obtained. In a second section, some properties of H∞ as a topological algebra are discussed. In particular, the spectrum is identified (under some rather restrictive conditions on the Banach space) and this is used to obtain a representation for strictly continuous homomorphisms between such H∞-algebras

Item Type: | Article |
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Uncontrolled Keywords: | Space of bounded holomorphic functions on the open unit ball of a Banach space; Strict topology; Approximation by polynomials; Continuous homomorphisms |

Subjects: | Sciences > Mathematics > Mathematical analysis |

ID Code: | 15314 |

References: | Alencar, R., ‘On the reflexivity and basis for P(mE)’, Proc. Roy. Irish Acad, 85 A (1985), 131–138. Alencar, R., Aron, R. and Dineen, S., ‘A reflexive space of holomorphic functions in infinitely many variables’, Proc. Amer. Math. Soc. 90 (1984), 407–411. Aron, R., ‘Compact polynomials and compact differentiable mappings between Banach spaces’, in Sém. Pierre Lelong, Lecture Notes in Maths 529 (Springer-Verlag, Heidelberg, Berlin, New York, 1976), pp. 213–222. Aron, R. and Berner, P., ‘A Hahn-Banach extension theorem for analytic mappings’, Bull. Soc. Math. France 106 (1978), 3–24. Bochnak, J. and Siciak, J., ‘Polynomials and multilinear mappings’, in Algebraic properties of classes of analytic functions Sem. Analytic Functions II, (R.C. Buck, Editor), 1957, pp. 175–188. Buck, R.C., ‘Bounded continuous functions on locally compact spaces’, Michigan Math. J. 5 (1958), 95–104. Chae, S.B., Holomorphy and calculus in Banach spaces (Marcel Dekker, New York, 1985). Cooper, J.B., ‘The strict topology and spaces with mixed topology’, Proc. Amer. Math. Soc. 30 (1971), 583–592. Davie, A. and Gamelin, T., ‘A theorem on polynomial-star topology’, Proc. Amer. Math. Soc. 106 (1989), 351–356. Dunford, N., ‘Uniformity in linear spaces’, Trans. Amer. Math. Soc. 44 (1938), 305–356. Garnett, J., Bounded analytic functions (Academic Press, New York, 1981). Isidro, J., ‘Topological duality on the functional space (Hb(U;F), τb)’, Proc. Roy. Irish Acad. 79A (1979), 115–130. Köthe, G., Topological vector spaces, Grundlehren Math. Wiss. 159 (Springer-Verlag, Berlin, Heidelberg, New York, 1969). Landau, E., ‘Abschätzung der Koeffzientensumme einer Potenzreihe’, Arch. Math. Phys. 21 (1913), 42–50. Prieto, A., La topología estricta en espacios de funciones holomorfas, Thesis (Universidad Complutense de Madrid, 1989). Prieto, A., ‘Strict and mixed topologies on function spaces’, Math. Nachr. 155 (1992), 289–293. Prieto, A., ‘Sur le lemme de Schwarz en dimension infinite’, C.R. Acad. Sci. Paris Ser. I 314 (1992), 741–742. Rubel, L. and Shields, A., ‘The space of bounded analytic functions on a region’, Ann. Inst. Fourier (Grenoble) 16 (1966), 235–277. Tsirelson, B., ‘Not every Banach space contains an imbedding of lp or co’, Funct. Anal. Appl. 8 (1974), 138–141. |

Deposited On: | 22 May 2012 09:08 |

Last Modified: | 06 Feb 2014 10:21 |

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