Two Theorems in Superconvex Analysis

Abstract

The present paper deals with the superconvex theory developed in the Thesis 1977 and in two subsequent papers 1980/81 of Rodé. A central result of it is a Dini type theorem: If on the superconvex space X an increasing sequence of superconvex functions f_n: X→ [−∞, ∞ [ with limit function f: X→[−∞, ∞[ is such that at each x∈X one has f_n (x) = f (x) for some n∈ℕ, then Inf f_n → Inf f. The condition that the limit function be attained at each point seems to be severe but is fulfilled quite often. However, it will be shown below that it is not essential: It suffices that there exists a sequence of positive numbers c_n ↓ 0 such that f (x) − f_n (x)=0(c_n ) for n→∞ at each x∈X. Rodé proved his result via a certain intersection theorem which resembles the classical Baire theorem. We retain this procedure and first extend the intersection theorem, to its ultimate limit as a counterexample reveals.