L ∞ Spaces of Vector-Valued Functions as Spaces of Continuous Functions

Abstract

It is proved that for any decomposable perfect measure space (𝑍, 𝒜, 𝜇), the space 𝐿𝜔∗∞ (𝜇, 𝐸*) of essentially bounded weak* measurable functions on 𝑍 to 𝐸* is linearly isometric to the space 𝐶(𝑍,𝐸∗*) of continuous functions on 𝑍 to 𝐸∗*, the latter space is being provided with the supremum norm ‖𝑔‖∞ = sup𝑧∈𝑍‖𝑔(𝑧)‖⁡ where 𝐸∗* stands for the space 𝐸* endowed with its weak* topology.

Keywords

• 𝐿 ∞ Space
• Vector-Valued Functions
• Perfect Measure
• Hyperstonean Space
• Continuous Function Spaces

Thanks

I am grateful to my teacher, the late Prof. Bahaettin Cengiz, who made a valuable contribution to this article.

References

1. E. Behrends, et al, L^p-structure in real Banach spaces, Lecture Notes in Mathematics, 613, Berlin-New-York, Springer-Verlag, 1977.
2. S. Bochner and R.E. Taylor, “Linear functionals on certain spaces of abstractly-valued functions”, Annals of Mathematics, 39(2), 913-944, 1938.
3. F. Bonsall and J. Duncan, Complete Normed Algebras, Berlin-Heidelberg-New York, Springer-Verlag, 1973.
4. M. Cambern and P. Greim, “The bidual of C(X,E)”, Proceedings of the American Mathematical Society, 85, 53-58, 1982.
5. M. Cambern and P. Greim, “The dual of a space of vector measures”, Mathematische Zeitschrift, 180, 373-378, 1982.
6. B. Cengiz, “On the duals of Lebesgue-Bochner L^p spaces”, Proceedings of the American Mathematical Society, 114, 923-926, 1992.
7. B. Cengiz, “The isometries of the Bochner space L^p (μ,H) ”, Turkish Journal of Mathematics, 23(3), 1999.
8. J. Diestel and J. J. Uhl Jr., Vector Measures, Mathematical Surveys and Monographs no.15, American Mathematical Society., Providence, Rhode Island, 1977.

9. Dinculeanu N., Vector Measures, New-York, Pergamon Press, 1967.
10. J. Dixmier, “Sur certains espaces considérés par M.H. Stone, Summa Brasiliensis Mathematicae, 2(11) , 151-182, 1951.
11. J. Dugundji, Topology, Boston, Allyn and Bacon Inc., 1966.
12. N. Dunford and J. T. Schwartz, Linear Operators, Part I, New York, Interscience, 1958.
13. N. E. Gretsky and J. J. Uhl Jr., “Bounded linear operators on Banach function spaces of vector-valued functions”, Transactions American Mathematical Society, 167, 263-277, 1972.
14. E. Hewitt , K. Stromberg, Real and Abstract Analysis, New York, Springer-Verlag, 1965.
15. L. Gilman and M. Jerison, Rings of Continuous Functions, Princeton-Toronto-London-Melbourne, D. Van Nostrand Company, 1960.
16. H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Berlin- Heidelberg-New York, Springer-Verlag, 1974.
17. M. A. Naimark, Normed Algebras, The Netherlands, Wolters-Noordhoff Publishing, 1972.
18. B. J. Pettis, “On integration in vector measures”, Transactions American Mathematical Society, 44, 277-304, 1938.
19. H. L. Royden, Real Analysis, 3rd ed., London-New York, Collier-Macmillan, 1988.
20. H. H. Schaefer, Banach Lattices and Positive Operators, Berlin-Heidelberg-New York, Springer-Verlag, 1974.
21. B. Güntürk, B. Cengiz, “On some properties of hyperstonean spaces”, Turkish Journal of Mathematics, 5(42), 2288-2295, 2018.