Norm attaining operators and norming functionals

Authors:
Russell G. Bilyeu and Paul W. Lewis

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

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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 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 -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.

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DOI:
https://doi.org/10.1090/S0002-9939-1982-0652451-7

Article copyright:
© Copyright 1982
American Mathematical Society