Norm attaining operators and norming functionals
HTML articles powered by AMS MathViewer
- by Russell G. Bilyeu and Paul W. Lewis PDF
- Proc. Amer. Math. Soc. 85 (1982), 245-250 Request permission
Abstract:
The question of whether a countably additive measure with values in a Banach space attains the diameter of its range was unresolved. In this paper an example is given of a countably additive vector measure, taking values in a $C(K)$ space, for which the diameter of the range is not attained. A property stronger than the attainment of the diameter, but which is possessed by many measures taking values in $L$-spaces, is shown to fail for infinite-dimensional measures into a space having smooth dual. As an application of the concept of norming functional (the existence of which is equivalent to the attainment of diameter), a characterization is given of the countably additive measures into space having smooth dual.References
- R. Anantharaman, On exposed points of the range of a vector measure, Vector and operator valued measures and applications (Proc. Sympos., Alta, Utah, 1972) Academic Press, New York, 1973, pp. 7–22. MR 0333111
- William D. L. Appling, Summability of real-valued set functions, Riv. Mat. Univ. Parma (2) 8 (1967), 77–100. MR 251187
- Russell G. Bilyeu and Paul W. Lewis, Orthogonality and the Hewitt-Yosida theorem in spaces of measures, Rocky Mountain J. Math. 7 (1977), no. 4, 629–638. MR 450499, DOI 10.1216/RMJ-1977-7-4-629
- J. K. Brooks and P. W. Lewis, Linear operators and vector measures, Trans. Amer. Math. Soc. 192 (1974), 139–162. MR 338821, DOI 10.1090/S0002-9947-1974-0338821-5
- Joseph Diestel, Geometry of Banach spaces—selected topics, Lecture Notes in Mathematics, Vol. 485, Springer-Verlag, Berlin-New York, 1975. MR 0461094
- 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
- N. Dinculeanu, Vector measures, Hochschulbücher für Mathematik, Band 64, VEB Deutscher Verlag der Wissenschaften, Berlin, 1966. MR 0206189 N. Dunford and J. T. Schwartz, Linear operators. Part I, Interscience, New York, 1958.
- R. C. James, Orthogonality in normed linear spaces, Duke Math. J. 12 (1945), 291–302. MR 12199
- Jerry Johnson and John Wolfe, Norm attaining operators, Studia Math. 65 (1979), no. 1, 7–19. MR 554537, DOI 10.4064/sm-65-1-7-19
- V. I. Rybakov, On the theorem of Bartle, Dunford and Schwartz on vector-valued measures, Mat. Zametki 7 (1970), 247–254 (Russian). MR 260971
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 85 (1982), 245-250
- MSC: Primary 28B05; Secondary 46G10
- DOI: https://doi.org/10.1090/S0002-9939-1982-0652451-7
- MathSciNet review: 652451