Strongly extreme points and the Radon-Nikodým property
HTML articles powered by AMS MathViewer
- by Zhibao Hu
- Proc. Amer. Math. Soc. 118 (1993), 1167-1171
- DOI: https://doi.org/10.1090/S0002-9939-1993-1152279-5
- PDF | Request permission
Abstract:
We prove that if $K$ is a bounded and convex subset of a Banach space $X$ and $x$ is a point in $K$, then $x$ is a strongly extreme point of $K$ if and only if $x$ is a strongly extreme point of ${\overline K ^{\ast }}$ which is the weak$^{{\ast }}$ closure of $K$ in ${X^{{\ast }{\ast }}}$. We also prove that a Banach space $X$ has the Radon-Nikodým property if and only if for any equivalent norm on $X$, the unit ball has a strongly extreme point.References
- Richard D. Bourgin, Geometric aspects of convex sets with the Radon-Nikodým property, Lecture Notes in Mathematics, vol. 993, Springer-Verlag, Berlin, 1983. MR 704815, DOI 10.1007/BFb0069321
- David W. Dean, The equation $L(E,\,X^{\ast \ast })=L(E,\,X)^{\ast \ast }$ and the principle of local reflexivity, Proc. Amer. Math. Soc. 40 (1973), 146–148. MR 324383, DOI 10.1090/S0002-9939-1973-0324383-X
- R. E. Huff and P. D. Morris, Geometric characterizations of the Radon-Nikodým property in Banach spaces, Studia Math. 56 (1976), no. 2, 157–164. MR 412776, DOI 10.4064/sm-56-2-157-164
- Ken Kunen and Haskell Rosenthal, Martingale proofs of some geometrical results in Banach space theory, Pacific J. Math. 100 (1982), no. 1, 153–175. MR 661446, DOI 10.2140/pjm.1982.100.153
- Peter Morris, Disappearance of extreme points, Proc. Amer. Math. Soc. 88 (1983), no. 2, 244–246. MR 695251, DOI 10.1090/S0002-9939-1983-0695251-5
Bibliographic Information
- © Copyright 1993 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 118 (1993), 1167-1171
- MSC: Primary 46B22
- DOI: https://doi.org/10.1090/S0002-9939-1993-1152279-5
- MathSciNet review: 1152279