A minimax theorem in infinite-dimensional topological vector spaces

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, $\phi :E\to \mathbf{R}$ a lower semicontinuous, convex and coercive functional, $I\subset \mathbf{R}$ a compact interval, with $0\in I$, $\psi :I\to \mathbf{R}$ a lower semicontinuous convex function. Then, for each $r>\phi (0)$, one has $$\underset{x\in X}{\sup }\underset{\lambda \in I}{\inf }(\phi (T(x)-\lambda x)+\psi (\lambda ))=r+\psi (0),$$ where $$X=\{x\in E:\phi (T(x))\le r\}.$$

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