Vol. 26, No. 3, 1968

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Normal structure in Banach spaces

Lawrence Peter Belluce, William A. Kirk and Eugene Francis Steiner

Vol. 26 (1968), No. 3, 433–440

A bounded convex subset K of a Banach space B has normal structure if for each convex subset H of K which contains more than one point there is a point x in H which is not a diametral point of H. The concept of normal structure (introduced by Brodskii and Milman) and a strengthening of this concept called complete normal structure have been of fundamental importance in some recent investigations concerned with determining conditions on weakly compact K under which the members of any commutative family 𝒯 of nonexpansive mappings of K into itself have a common fixed-point. A more thorough study of these concepts is initiated in the present paper. The theorems obtained primarily concern product spaces composed of spaces which possess normal structure.

Mathematical Subject Classification
Primary: 46.10
Received: 11 October 1966
Published: 1 September 1968
Lawrence Peter Belluce
William A. Kirk
Eugene Francis Steiner