A minimax theorem in infinite-dimensional topological vector spaces

2015/08/20 by Biagio Ricceri, Ricceri, Biagio
Computer Science · Mathematics · #Advanced Banach Space Theory #FOS: Mathematics #Fixed Point Theorems Analysis #Functional Analysis (math.FA) #Optimization and Variational Analysis #math.FA

paper · pdf · doi:10.48550/arxiv.1508.04891

openalex publication_date 2015/08/20 · arxiv created 2015/09/08 · arxiv updated 2015/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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→ E a non-zero compact linear operator, φ:E→ \bf R a lower semicontinuous, convex and coercive functional, I⊂ \bf R a compact interval, with 0∈ I, ψ:I→ \bf R a lower semicontinuous convex function. Then, for each r>φ(0), one has sup_{x∈ X}inf_{λ∈ I}(φ(T(x)-λx)+ψ(λ))=r+ψ(0) , where X=\x∈ E : φ(T(x))≤ r\ .

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