Super weakly compact subsets of Banach spaces.

Qing-Jin Cheng Xiamen University, China

Abstract: We introduce a concept of super weak compactness for a bounded weak closed subset in Banach spaces. The class of super weakly compact sets in Banach spaces lies between compact and the weakly compact and shares mangy good properties of those classes. In my lecture, we will consider the following several closely related topics: i)

Characterize the super weakly compact sets in general Banach spaces, and the criteria of super weakly compact sets in some special spaces.

ii)

Topological properties of super weakly compact sets.

iii)

Properties of Banach spaces which are generated by super weakly compact sets.

Some main references of my lecture: [1] Amir and J. Lindenstrauss, The structure of weakly compact sets in Banach spaces, Annals of Mathematics, 88 (1968), 35-44. [2] Y. Benyamini, T. Starbird, Embedding weakly compact sets into Hilbert space, Israel J. Math. 23 (1976), no. 2, 137–141. [3] L. Cheng, Q. Cheng, B. Wang, W. Zhang, On super-weakly compact sets and uniformly convexifiable sets, Studia. Math. 199 (2010), no. 2, 145-169. [4] J. Lindenstrauss, Weakly compact sets—their topological properties and the Banach spaces they generate, Symposium on Infinite-Dimensional Topology (Louisiana State Univ., Baton Rouge, La., 1967), Princeton Univ. Press, Princeton, N. J., 1972, pp. 235–273. Ann. of Math. Studies, No. 69. [5] M. Raja, Finitely dentable functions, operators and sets, J. Convex Anal.15 (2008), no. 2, 219–233. [6] F. Albiac, N. j. Kalton, Topics in Banach space Theory. [7] M. Fabian, P. Habala, P. Hajek, V. Montesinos, J. Pelant, and V. Zizler, Functional analysis and in¯nite dimensional geometry, Canad. Math. Soc. Books in Mathematics 8, Springer-Verlag, New York, 2001.