On exposed points of the range of a vector measure. II

Author:
R. Anantharaman

Journal:
Proc. Amer. Math. Soc. **55** (1976), 334-338

MathSciNet review:
0399851

Full-text PDF Free Access

Abstract | References | Additional Information

Abstract: If a weakly compact convex set in a real Banach space is strongly exposed by a dense set of functionals in , it is proved that the functionals which expose form a residual set in . If is a measure, it follows that the set of exposing functionals of its range is a residual in . This, in turn, is found to be equivalent to a theorem of B. Walsh on the residuality of functionals for which .

If the set of exposed points of is weakly closed and is the restriction of to any set , it is further proved that every exposed point of the range of is of the form , where and is an exposed point of .

**[1]**D. Amir and J. Lindenstrauss,*The structure of weakly compact sets in Banach spaces*, Ann. of Math. (2)**88**(1968), 35–46. MR**0228983****[2]**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****[3]**R. G. Bartle, N. Dunford, and J. Schwartz,*Weak compactness and vector measures*, Canad. J. Math.**7**(1955), 289–305. MR**0070050****[4]**T. Husain and I. Tweddle,*On the extreme points of the sum of two compact convex sets*, Math. Ann.**188**(1970), 113–122. MR**0271688****[5]**Joram Lindenstrauss,*On operators which attain their norm*, Israel J. Math.**1**(1963), 139–148. MR**0160094****[6]**Neil W. Rickert,*Measures whose range is a ball*, Pacific J. Math.**23**(1967), 361–371. MR**0222244****[7]**V. I. Rybakov,*On the theorem of Bartle, Dunford and Schwartz on vector-valued measures*, Mat. Zametki**7**(1970), 247–254 (Russian). MR**0260971****[8]**S. L. Troyanski,*On locally uniformly convex and differentiable norms in certain non-separable Banach spaces*, Studia Math.**37**(1970/71), 173–180. MR**0306873****[9]**Bertram Walsh,*Mutual absolute continuity of sets of measures*, Proc. Amer. Math. Soc.**29**(1971), 506–510. MR**0279275**, 10.1090/S0002-9939-1971-0279275-X

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1976-0399851-8

Keywords:
Exposed points,
exposing functionals,
range of vector measures,
weakly compact convex sets

Article copyright:
© Copyright 1976
American Mathematical Society