Topological semilattices with path-connected intervals are special abstract convex spaces. In this paper, we obtain generalized KKM type theorems and their analytic formulations, maximal element theorems and collectively fixed point theorems on abstract convex spaces. We also apply them to topological semilattices with path-connected intervals, and obtain generalized forms of the results of Hor...

We investigate certain envelopes of open sets in dual Banach spaces which are related to extending holomorphic functions. We give a variety of examples of absolutely convex sets showing that the extension is in many cases not possible. We also establish connections to the study of iterated weak* sequential closures of convex sets in the dual of separable spaces.

Let X be a normed linear space, and let S(X) = fx 2 X : kxk = 1g be the unit sphere of X. Brodskii and Milman 1] introduced the following geometric concept: Deenition 1. A bounded, convex subset K of a Banach space X is said to have normal structure if every convex subset H of K that contains more than one point contains a point x 0 2 H, such that supfkx 0 ? yk : y 2 Hg < d(H); where d(H) = sup...

In this note, we investigate non-locally-convex topological vector spaces for which the closed graph theorem holds. In doing so, we introduce new classes of topological vector spaces. Our study includes a direct extension of Pták duality to the non-locally-convex situation.