A minimax theorem in infinite-dimensional topological vector spaces

TL;DR

This paper establishes a minimax theorem in infinite-dimensional topological vector spaces and applies it to derive a specific equality involving a reflexive Banach space, a compact operator, and convex functionals.

Contribution

The paper introduces a new minimax theorem in infinite-dimensional spaces and demonstrates its application to a class of convex optimization problems.

Findings

Proves a minimax theorem in infinite-dimensional topological vector spaces.

Derives an equality involving convex functionals, a compact operator, and a reflexive Banach space.

Provides conditions under which the supremum and infimum of a functional expression are equal.

Abstract

In this paper, we obtain a minimax theorem by means of which, in turn, we prove the following result: Let $E$ be an infinite-dimensional reflexive real Banach space, $T:E\to E$ a non-zero compact linear operator, $\varphi:E\to {\bf R}$ a lower semicontinuous, convex and coercive functional, $I\subset {\bf R}$ a compact interval, with $0\in I$, $\psi:I\to {\bf R}$ a lower semicontinuous convex function. Then, for each $r>\varphi(0)$, one has $$\sup_{x\in X}\inf_{\lambda\in I}(\varphi(T(x)-\lambda x)+\psi(\lambda))=r+\psi(0)\ ,$$ where $$X=\{x\in E : \varphi(T(x))\leq r\}\ .$$