Hostname: page-component-745bb68f8f-mzp66 Total loading time: 0 Render date: 2025-01-29T19:57:59.743Z Has data issue: false hasContentIssue false
  • English
  • Français

Biduals of weighted banach spaces of analytic functions

Part of: Normed linear spaces and Banach spaces; Banach lattices Topological linear spaces and related structures Linear function spaces and their duals

Published online by Cambridge University Press:  09 April 2009

K. D. Bierstedt  and
W. H. Summers
K. D. Bierstedt
Affiliation:
FB 17, MathematikUniversität-GH-PaderbornPostfach 16 21 D-4790 Paderborn, Germany
W. H. Summers
Affiliation:
Department of Mathematical Sciences University of ArkansasFayetteville, AR 72701, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For a positive continuous weight function ν on an open subset G of CN, let Hv(G) and Hv0(G) denote the Banach spaces (under the weighted supremum norm) of all holomorphic functions f on G such that ν f is bounded and ν f vanishes at infinity, respectively. We address the biduality problem as to when Hν(G) is naturally isometrically isomorphic to Hν0(G)**, and show in particular that this is the case whenever the closed unit ball in Hν0(G) in compact-open dense in the closed unit ball of Hν(G).

MSC classification

Secondary: 46E15: Banach spaces of continuous, differentiable or analytic functions 46A70: Saks spaces and their duals (strict topologies, mixed topologies, two-norm spaces, co-Saks spaces, etc.) 46B10: Duality and reflexivity
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 54 , Issue 1 , February 1993 , pp. 70 - 79
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Anderson, J. M. and Duncan, J., ‘Duals of Banach spaces of entire functions’ Glasgow Math. J. 32 (1990), 215–220.CrossRefGoogle Scholar
[2]Ando, T., ‘On the predual of H∞’, Comment. Math. Prace Mat. (1978), 33–40.Google Scholar
[3]Bierstedt, K. D. and Bonet, J., ‘Biduality in Fréchet and (LB)-spaces’, in: Progress in Functional Analysis, North-Holland Math. Stud. (North-Holland, Amsterdam) pp. 113–133, to appear.Google Scholar
[4] Bierstedt, K. D., Bonet, J. and Galbis, A., ‘Weighted spaces of holomorphic functions on balanced domains’, Michigan Math. J., to appear.Google Scholar
[5]Bierstedt, K. D., Meise, R. and Summers, W. H., ‘A projective description of weighted inductive limits’, Trans. Amer. Math. Soc. 272 (1982), 107–160.Google Scholar
[6]Cooper, J. B., Saks spaces and applications to functional analysis, North-Holland Math. Stud. 28 (North-Holland, Amsterdam, 1978).CrossRefGoogle Scholar
[7]Gamelin, T. W., Uniform algebras (Prentice-Hall, Englewood Cliffs, 1969).Google Scholar
[8]Ng, K.-F., ‘On a theorem of Dixmier’ Math. Scand. 29 (1971), 279–280.CrossRefGoogle Scholar
[9]Rubel, L. A. and Ryff, J. V., ‘The bounded weak-star topology and the bounded analytic functions’, J. Funct. Anal. 5 (1970), 167–183.CrossRefGoogle Scholar
[10]Rubel, L. A. and Shields, A. L., ‘The space of bounded analytic functions on a region’, Ann. Inst. Fourier (Grenoble) 16 (1966), 235–277.CrossRefGoogle Scholar
[11]Rubel, L. A. and Shields, A. L., ‘The second duals of certain spaces of analytic functions’, J. Austral. Math. Soc. 11 (1970), 276–280.Google Scholar
[12]Shapiro, J. H., ‘Weak topologies on subspaces of C(S)’, Trans. Amer. Math. Soc. 157 (1971), 471–479.Google Scholar
[13]Shapiro, J. H., ‘The bounded weak star topological and the general strict topology’, J. Funct. Anal. 8 (1971), 275–286.CrossRefGoogle Scholar
[14]Shapiro, J. H., Shields, A. L. and Taylor, G. D., ‘The second duals of some function spaces’, unpublished manuscript, 1970/71.Google Scholar
[15]Summers, W. H., ‘Dual spaces of weighted spaces’, Trans. Amer. Math. Soc. 151 (1970), 323–333.CrossRefGoogle Scholar
[16]Waelbroeck, L., ‘Duality and the injective tensor product’, Math. Ann. 163 (1966), 122–126.Google Scholar
[17]Williams, D. L., Some Banach spaces of entire functions (Ph. D. Thesis, University of Michigan, 1967).Google Scholar
[18]Wojtaszczyk, P., Banach spaces for analysts (Cambridge Univ. Press, Cambridge, 1990).Google Scholar