Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-04T21:23:18.596Z Has data issue: false hasContentIssue false
  • English
  • Français

Dual functors and the Radon-Nikodym property in the category of Banach spaces

Part of: Methods of category theory in functional analysis

Published online by Cambridge University Press:  09 April 2009

John Wick Pelletier
John Wick Pelletier
Affiliation:
Department of Mathematics York University, Keele Street Downsview, Ontario, Canada
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.

The notion of duality of functors is used to study and characterize spaces satisfying the Radon-Nikodym property. A theorem of equivalences concerning the Radon-Nikodym property is proved by categorical means; the classical Dunford-Pettis theorem is then deduced using an adjointness argument. The functorial properties of integral operators, compact operators, and weakly compact operators are discussed. It is shown that as an instance of Kan extension the weakly compact operators can be expressed as a certain direct limit of ordinary hom functors. Characterizations of spaces satisfying the Radon-Nikodym property are then given in terms of the agreement of dual functors of the functors mentioned above.

MSC classification

Secondary: 46M15: Categories, functors
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 27 , Issue 4 , June 1979 , pp. 479 - 494
Copyright
Copyright © Australian Mathematical Society 1979

References

Cigler, J. (1976), Tensor products of functors on categories of Banach spaces (Springer Lecture Notes 540, 164–187).Google Scholar
Davis, W. J., Figiel, T., Johnson, W. B. and Pelczynski, A. (1974), ‘Factoring weakly compact operators’, J. Functional Analysis 17, 311–327.CrossRefGoogle Scholar
Day, B. J. and Kelly, G. M. (1969) Enriched functor categories (Springer Lecture Notes 106, 178–191.)Google Scholar
Diestel, J. (1972), ‘The Radon-Nikodym property and the coincidence of integral and nuclear operators’, Rev. Roumaine Math. Pures Appl. 17, 1611–1620.Google Scholar
Diestel, J. (1975), Geometry of Banach spaces—selected topics (Springer Lecture Notes 485).Google Scholar
Diestel, J. and Uhl, J. J. Jr, (1976), ‘The Radon-Nikodym Theorem for Banach space valued measures’, Rocky Mountain J. Math. 6, 1–46.Google Scholar
Dubuc, E. J. (1970), Kan extensions in enriched Category theory (Springer Lecture Notes 145).Google Scholar
Grothendieck, A. (1966), Produits tensoriels topologiques et espaces nucléaires (Mem. Amer. Math. Soc. 16).Google Scholar
Herz, C. and Pelletier, J. Wick (1976), ‘Dual functors and integral operators in the category of Banach spaces’, J. Pure Appl. Algebra 8, 5–22.CrossRefGoogle Scholar
Losert, V. (1976). Dualität von Funktoren und Operatorenideale (Dissertation, Universität Wien).Google Scholar
Linton, F. E. J. (1965), ‘Autonomous categories and duality of functors’, J. Algebra 2, 315–349.Google Scholar
Lotz, H. (1971), Topological tensor products, linear mappings and nuclear spaces (Lecture Notes, University of Illinois).Google Scholar
MacLane, S. (1971), Categories for the working mathematician (Springer-Verlag, New York).Google Scholar
Mityagin, B. S. and Švarc, A. S. (1964), ‘Functors in categories of Banach spaces’, Russian Math. Surveys 19 (2), 65–127.Google Scholar
Persson, A. and Pietsch, A. (1969), ‘p-Nukleare und p-integrale Abbildungen’, Studia Math. 33, 19–62.Google Scholar
Phillips, R. S. (1943), ‘On weakly compact subsets of a Banach space’, Amer. J. Math. 65, 108–136.Google Scholar