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

Author:
R. Anantharaman

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

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

MathSciNet review:
0399851

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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**37**#4562. MR**0228983 (37:4562)****[2]**R. Anantharaman,*On exposed points of the range of a vector measure*, Proc. Conf. on Vector Measures (Snowbird, Utah), Academic Press, New York, 1973, pp. 7-22. MR**0333111 (48:11436)****[3]**R. G. Bartle, N. Dunford and J. T. Schwartz,*Weak compactness and vector measures*, Canad. J. Math.**7**(1955), 289-305. MR**16**, 1123. MR**0070050 (16:1123c)****[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**42**#6571. MR**0271688 (42:6571)****[5]**J. Lindenstrauss,*On operators which attain their norm*, Israel J. Math.**1**(1963), 139-148. MR**28**#3308. MR**0160094 (28:3308)****[6]**N. W. Rickert,*Measures whose range is a ball*, Pacific J. Math.**23**(1967), 361-371. MR**36**#5296. MR**0222244 (36:5296)****[7]**V. I. Rybakov,*Theorem of Bartle, Dunford and Schwartz concerning vector measures*, Mat. Zametki**7**(1970), 247-254 = Math. Notes**7**(1970), 147-151. MR**41**#5591. MR**0260971 (41:5591)****[8]**S. L. Trojanski,*On locally uniformly convex and differentiable norms in certain non-separable Banach spaces*, Studia Math.**37**(1970/71), 173-180. MR**46**#5995. MR**0306873 (46:5995)****[9]**B. J. Walsh,*Mutual absolute continuity of sets of measures*, Proc. Amer. Math. Soc.**29**(1971), 506-510. MR**43**#4998. MR**0279275 (43:4998)**

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