强严格凸和中点局部一致凸

Strongly Strict Convexity and Locally Uniform Convexity

证明了Ｂａｎａｃｈ空间Ｘ是局部一致非方的当且仅当对任意ｘ∈Ｓ（Ｘ），都有ｄＸ（ｘ，１）＞０；Ｘ是强严格凸的当且仅当对任意ｘ∈Ｓ（Ｘ），ｙｎ∈Ｓ（Ｘ）和α∈Ｒ，若‖ｘ＋αｙｎ‖→１和‖ｘ－αｙｎ‖→１，则α＝０；

It is shown that Banach space X is locally uniformly non square if and only if d X(x,1)>0 for all x∈S(X) ; X is strongly strictly convex if and only if whenver for any x∈S(X),y n∈S(X) and α∈R, if ‖x+αy n‖→0 , and ‖x-αy n‖→1, then α=0 ;and X is strongly strictly convex if and only if X is midpoint locally uniformly convex.