A minimax theorem for Lindelöf sets (Q1730803)

Let $\mathcal{X}$ a real vector space and $\{f_i: \mathcal{X} \rightarrow \mathbb{R}\}_{i\in I}$ be a family of real functionals on $\mathcal{X}$. A subset $C$ of $\mathcal{X}$ is said to be upper closed for $\{ f_i \}_{i\in I}$ if, for any sequence $\{x_n\}$ in $C$ there exists $x^* \in C$ such that for any $i \in I$, $\limsup_{n \rightarrow \infty} f_i(x_n) \leq f_i(x^*)$. The main result of the paper is the following minimax theorem: \par Let $X$ be a nonempty convex subset of a vector space $\mathcal{X}$ and $Y$ be a nonempty Lindelöf convex subset of a Hausdorff topological space. Let $f: X \times Y \rightarrow \mathbb{R}$ be function that satisfies the following conditions: (i) there exists $M > 0$ such that for any $x \in X$ and $y \in Y$, $| f(x,y)| \leq M$ (ii) for each $y\in Y$, $f(\cdot, y)$ is convex; (iii) for each $x \in X$, $f(x, \cdot)$ is concave and upper semicontinuous; (iv) $Y$ is upper closed for $\{f(x, \cdot)\}_{x\in X}$. Then, $\inf_{x\in X} \sup_{y\in Y} f(x,y) = \sup_{y\in Y} \inf_{x\in X} f(x,y).$