I was struggling a little bit with the following question.
Let $A$ and $B$ be two non-empty sets and $f:A \times B \rightarrow \mathbb{R}$. Then it holds $\sup_{y\in Y}\left(\inf_{x\in X}f(x,y)\right)\leq \inf_{x\in X}\left(\sup_{y\in Y}f(x,y)\right)$.
Now, what I was wondering about is when exactly we have equality above, i.e. $\sup_{y\in Y}\left(\inf_{x\in X}f(x,y)\right)= \inf_{x\in X}\left(\sup_{y\in Y}f(x,y)\right)$.
Are there any theorems that give us conditions under which this holds?
I would appreciate your help.