Approximate Convexity and Submonotonicity in Locally Convex Spaces

approximate convexity and submonotonicity in locally convex spaces

Approximate convexity and submonotonicity

It is shown that a locally Lipschitz function is approximately convex if, and only if, its Clarke subdifferential is a submonotone operator. Consequently, in finite dimensions, the class of locally Lipschitz approximately convex functions coincides with the class of lower-C functions. Directional approximate convexity is introduced and shown to be a natural extension of the class of lower-C fun...

Asymmetric locally convex spaces

The aim of the present paper is to introduce the asymmetric locally convex spaces and to prove some basic properties. Among these I do mention the analogs of the EidelheitTuckey separation theorems, of the Alaoglu-Bourbaki theorem on the weak compactness of the polar of a neighborhood of 0, and a Krein-Milman-type theorem. These results extend those obtained by Garcı́a-Raffi et al. (2003) and Co...

Seminorms and Locally Convex Spaces

The first point is to describe vector spaces with topologies arising from (separating) families of semi-norms. These all turn out to be locally convex, for straightforward reasons. The second point is to check that any locally convex topological vectorspace's topology can be given by a collection of seminorms. These seminorms are made in a natural way from a local basis consisting of balanced c...

Convexity Conditions for Non - Locally Convex Lattices

for any x 1 ( . . . , x,, GX. A theorem of Aolci and Rolewicz (see [18]) asserts that if in (1.3) C = 2~\ then X is p-normable. We can then equivalently re-norm X so that in (1.4) JB = 1. If in addition X is a vector lattice and ||x||<||y|| whenever |x|<|y| we say that X is a quasi-Banach lattice. As in the case of Banach lattices [13] we may make the following definitions. We shall say that X ...

On the dual of certain locally convex function spaces

In this paper, we first introduce some function spaces, with certain locally convex topologies, closely  related to the space of real-valued continuous functions on $X$,  where $X$ is a $C$-distinguished topological space. Then, we show that their dual spaces can be identified in a natural way with certain spaces of Radon measures.