we first obtain some properties of a fundamentally nonexpansive self-mapping on a nonempty subset of a banach space and next show that if the banach space is having the opial condition, then the fixed points set of such a mapping with the convex range is nonempty. in particular, we establish that if the banach space is uniformly convex, and the range of such a mapping is bounded, closed and con...

In this paper, a Krein-Milman  type theorem in $T_0$ semitopological cone is proved,  in general. In fact, it is shown that in any locally convex $T_0$ semitopological cone, every convex compact saturated subset is the compact saturated convex hull of its extreme points, which improves the results of Larrecq.

The present paper is concerned with Lipschitz properties of convex mappings. One considers the general context of mappings defined on an open convex subset Ω of a locally convex space X and taking values in a locally convex space Y ordered by a normal cone. One proves also equi-Lipschitz properties for pointwise bounded families of continuous convex mappings, provided the source space X is barr...

Let F be an arbitrary topological vector space; we shall say that a subset S of F is quasi-convex if the set of continuous affine functionals on 5 separates the points of S. If X is a Banach space and T: X -* F is a continuous linear operator, then T is quasi-convex if T(U) is quasiconvex, where U is the unit ball of X. In the case when T is compact, T(U) is quasi-convex if and only if it is af...

These notes give a summary of results that everyone who does work in functional analysis should know about the weak topology on locally convex topological vector spaces and the weak-* topology on their dual spaces. The most striking of the results we prove is Theorem 9, which shows that a subset of a locally convex space is bounded if and only if it is weakly bounded. It is straightforward to p...

1. N o t a t i o n a n d pre l iminary ideas . A sequence space is a vector subspace of the space co of all real (or complex) sequences. A sequence space E with a locally convex topology r is called a Kspace if the inclusion map E —* co is continuous, when co is endowed with the product topology (co = II^Li (R)*). A i£-space E with a Frechet (i.e., complete, metrizable and locally convex) topol...

We first obtain some properties of a fundamentally nonexpansive self-mapping on a nonempty subset of a Banach space and next show that if the Banach space is having the Opial condition, then the fixed points set of such a mapping with the convex range is nonempty. In particular, we establish that if the Banach space is uniformly convex, and the range of such a mapping is bounded, closed and con...

Let X be a completely regular Hausdorff space, let V be a system of weights on X and let T be a locally convex Hausdorff topological vector space. Then CVb(X, T) is a locally convex space of vector-valued continuous functions with a topology generated by seminorms which are weighted analogues of the supremum norm. In the present paper we characterize multiplication operators on the space CVb(X,...