Geodesic
Spaces of Continuous Vector Functions as Duals
Canadian mathematical bulletin, Tome 31 (1988) no. 1, pp. 70-78

Voir la notice de l'article provenant de la source Cambridge University Press

A well known result due to Dixmier and Grothendieck for spaces of continuous scalar-valued functions C(X), X compact Hausdorff, is that C(X) is a Banach dual if, and only if, Xis hyperstonean. Moreover, for hyperstonean X, the predual of C(X) is strongly unique. Here we obtain a formulation of this result for spaces of continuous vector-valued functions. It is shown that if E is a Hilbert space and C(X, (E, σ *) ) denotes the space of continuous functions on X to E when E is provided with its weak * ( = weak) topology, then C(X, (E, σ *) ) is a Banach dual if, and only if, X is hyperstonean. Moreover, for hyperstonean X, the predual of C(X, (E, σ *) ) is strongly unique.
DOI : 10.4153/CMB-1988-011-3
Mots-clés : 46E40, 46E15, 46G10 
Cambern, Michael; Greim, Peter. Spaces of Continuous Vector Functions as Duals. Canadian mathematical bulletin, Tome 31 (1988) no. 1, pp. 70-78. doi: 10.4153/CMB-1988-011-3
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[1] 1. Behrends, E., et al., L -structure in real Banach spaces, Lecture Notes in Mathematics 613, Springer-Verlag, Berlin-Heidelberg-New York, 1977. Google Scholar

[2] 2. Cambern, M. and Greim, P., The bidual of C(X, E), Proc. Amer. Math. Soc. 85 (1982), pp. 53–58. Google Scholar

[3] 3. Cambern, M. and Greim, P., The dual of a space of vector measures, Math. Z. 180 (1982), pp. 373–378. Google Scholar

[4] 4. Cembranos, P., C(K, E) contains a complemented copy of c§, Proc. Amer. Math. Soc. 91 (1984), pp. 556–558. Google Scholar

[5] 5. Day, M. M., Normed linear spaces, 3rd ed., Springer-Verlag, Berlin-Heidelberg-New York, 1973. Google Scholar

[6] 6. Diestel, J. and Uhl, J. J. Jr., Vector measures, Math. Surveys 15, Amer. Math. Soc, Providence, R.I., 1977. Google Scholar

[7] 7. Dinculeanu, N., Vector measures, Pergamon Press, New York, 1967. Google Scholar

[8] 8. Dixmier, J., Sur certains espaces considérés par Stone M. H., Summa Brasil. Math. 2 (1951), pp. 151–182. Google Scholar

[9] 9. Dunford, N. and Schwartz, J. T., Linear operators, Part I, Interscience, New York, 1958. Google Scholar

[10] 10. Greim, P., Banach spaces with the Lx∼ Banach-S tone property, Trans. Amer. Math. Soc. 287 (1985), pp. 819–828. Google Scholar

[11] 11. Grothendieck, A., Une caractérisation vectorielle métrique des espaces L1 , Canadian J. Math. 7 (1955), pp. 552–561. Google Scholar

[12] 12. Holmes, R. B., Geometric functional analysis and its applications, Springer-Verlag, Berlin- Heidelberg-New York, 1975. Google Scholar

[13] 13. Lacey, H. E., The isometrical theory of classical Banach spaces, Springer-Verlag, Berlin- Heidelberg-New York, 1974. Google Scholar

[14] 14. Singer, I., Linear functional on the space of continuous mappings of a compact space into a Banach space, Rev. Roumaine Math. Pures Appl. 2 (1957), pp. 301–315. (Russian) Google Scholar

[15] 15. Taylor, A. E., Introduction to functional analysis, John Wiley and Sons, New York, 1958. Google Scholar

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