The bidual of $C(X, E)$
HTML articles powered by AMS MathViewer
- by Michael Cambern and Peter Greim PDF
- Proc. Amer. Math. Soc. 85 (1982), 53-58 Request permission
Abstract:
In this article we obtain, under certain conditions, a characterization of the bidual of the space $C(X,E)$ of continuous functions on a compact Hausdorff space $X$ to a Banach space $E$. It is shown that if $X$ is dispersed and $E$ is arbitrary, or if $X$ is arbitrary and ${E^* }$ has the Radon-Nikodym property, then $C{(X,E)^{**}}$ can be represented as a space of continuous functions on a compact Hausdorff space $Z$ to ${E^{**}}$ when the latter space is given its weak* topology.References
- Richard Arens, Operations induced in function classes, Monatsh. Math. 55 (1951), 1–19. MR 44109, DOI 10.1007/BF01300644
- S. D. Chatterji, Martingale convergence and the Radon-Nikodym theorem in Banach spaces, Math. Scand. 22 (1968), 21–41. MR 246341, DOI 10.7146/math.scand.a-10868
- J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964, DOI 10.1090/surv/015
- N. Dinculeanu, Vector measures, Hochschulbücher für Mathematik, Band 64, VEB Deutscher Verlag der Wissenschaften, Berlin, 1966. MR 0206189
- Hugh Gordon, The maximal ideal space of a ring of measurable functions, Amer. J. Math. 88 (1966), 827–843. MR 201961, DOI 10.2307/2373081
- Paul R. Halmos, Measure Theory, D. Van Nostrand Co., Inc., New York, N. Y., 1950. MR 0033869, DOI 10.1007/978-1-4684-9440-2
- Shizuo Kakutani, Concrete representation of abstract $(M)$-spaces. (A characterization of the space of continuous functions.), Ann. of Math. (2) 42 (1941), 994–1024. MR 5778, DOI 10.2307/1968778
- Samuel Kaplan, On the second dual of the space of continuous functions, Trans. Amer. Math. Soc. 86 (1957), 70–90. MR 90774, DOI 10.1090/S0002-9947-1957-0090774-3
- H. Elton Lacey, The isometric theory of classical Banach spaces, Die Grundlehren der mathematischen Wissenschaften, Band 208, Springer-Verlag, New York-Heidelberg, 1974. MR 0493279, DOI 10.1007/978-3-642-65762-7
- Ivan Zinger, Linear functionals on the space of continuous mappings of a compact Hausdorff space into a Banach space, Rev. Math. Pures Appl. 2 (1957), 301–315 (Russian). MR 96964
- Ivan Singer, Best approximation in normed linear spaces by elements of linear subspaces, Die Grundlehren der mathematischen Wissenschaften, Band 171, Publishing House of the Academy of the Socialist Republic of Romania, Bucharest; Springer-Verlag, New York-Berlin, 1970. Translated from the Romanian by Radu Georgescu. MR 0270044, DOI 10.1007/978-3-662-41583-2
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 85 (1982), 53-58
- MSC: Primary 46E40; Secondary 46B22
- DOI: https://doi.org/10.1090/S0002-9939-1982-0647896-5
- MathSciNet review: 647896