Sion's minimax theorem is stated as:
Let $X$ be a compact convex subset of a linear topological space and $Y$ a convex
subset of a linear topological space. Let $f$ be a real-valued function on $X \times Y$
such that 1. $f(x, \cdot)$ is
upper semicontinuous and quasi-concave on $Y$ for each $x \in X$.
2. $f( \cdot, y)$is lower semicontinuous and quasi-convex on $X$ for each $y \in Y$. Then: $$\inf_{x \in X}\sup_{y \in Y}f(x,y) = \sup_{y \in Y}\inf_{x \in X}f(x,y)$$
If we drop the condition of compactness of $X$, that is, both $X$ and $Y$ are only convex subsets of two linear topological spaces, is there an example such that $$\inf_{x \in X}\sup_{y \in Y}f(x,y) > \sup_{y \in Y}\inf_{x \in X}f(x,y)$$
Added: In this thread, Seyhmus Güngören propose a question on Von Neumann's minimax theorem which can be sharpened by Sion's minimax theorem as pointed out in its comments. This elementary proof of Sion's minimax theorem seems to rely crucially on the compactness of $X$. So a natural follow-up question is what's the example that prevent minimax theorem to drop compactness totally in this setting?
Best Answer
Let $f(x,y) = x + y$ on ${\mathbb R} \times {\mathbb R}$. Then $\inf_{x \in X} \sup_{y \in Y} f(x,y) = + \infty$ but $\sup_{y \in Y} \inf_{x \in X} f(x,y) = -\infty$.