Tom Richmond* ([email protected]). Complementation in the Lattice of Locally Convex Topologies. We find all locally convex homogeneous topologies on (R,≤) and determine which of these have locally convex complements. Among the locally convex topologies on a n-point totally ordered set, each has a locally convex complement, and at least n of them have 2n−1 locally convex complements. For any ...

A convex subset B of a real locally convex space X is said to have the separation property if it can be separated from any closed convex subset A of X, which is disjoint from B, by a closed hyperplane. The strong separation theorem says that if B is weakly compact then it has the separation property. In this paper, we present several versions for the converse and discuss some applications. For ...

For convex continuous functions f, g defined respectively in neighborhoods of points x, y in a normed linear space, a formula for the distance between ∂f(x) and ∂g(y) in terms of f, g (i.e. without using the dual) is proved. Some corollaries, like a new characterization of the subdifferential of a continuous convex function at a point, are given. This, together with a theorem from [4], implies ...

A theory of differential calculus for nonlinear maps between general locally convex spaces is developed. All convergence notions are topological, and only familiarity with basic results from point set topology, differential calculus in Banach spaces, and locally convex space theory is assumed. The chain rule for continuous kth order differentiability, smoothness of inverse functions, and functi...

We provide examples of non-locally compact geodesic Ptolemy metric spaces which are not uniquely geodesic. On the other hand, we show that locally compact, geodesic Ptolemy metric spaces are uniquely geodesic. Moreover, we prove that a metric space is CAT(0) if and only if it is Busemann convex and Ptolemy.

A subset M of a topological space 5 is said to have a convex metric (even though S may have no metric) if the subspace M of 5 has a convex metric. It is known [5 J that a compact continuum is locally connected if it has a convex metric. The question has been raised [5] as to whether or not a compact locally connected continuum M can be assigned a convex metric. Menger showed [5] that M is conve...

where g:(0, 1)-»X is an essentially bounded strongly measurable function. In this paper we examine analogues of the Radon-Nikodym Property for quasiBanach spaces. If 0 < p < 1, there are several possible ways of defining "differentiable" operators on Lp, but they inevitably lead to the conclusion that the only differentiable operator is zero. For example, a differentiable operator on L\ has the...