On boundedly-convex functions on pseudo-topological vector spaces

Abstract

Notions of a boundedly convex function and of a Lipschitz-continuous function are extended to the case of functions on pseudo-topological vector spaces. It is proved that for “good” pseudo-topologizers Ψ, any continuous Ψ-boundedly convex function is Ψ-differentiable and its derivative is Ψ-Lipschitz-continuous. As a corollary, it is shown that any boundedly convex function is Hyers-Lang differentiable.

Keywords

• Filter
• pseudo-topology
• convergence space
• pseudo-topologizer
• pseudo-topological vector space
• boundedly-convex
• Lipschitz-continuous.