Are there some common ways to approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets? (note that the unit ball isn't compact.)
My goal is to prove a statement which holds for a compact set, also holds on the unit ball. The proof for a compact set makes use of the definition of compactness (i.e. an open cover contains a finite open cover).
I also wonder what are some general approaches for proving such generalization?
Thanks.
My specific problem is to show that:
given a function $f: X \times Y \to \mathbb R$, if we can show that
when $X$ is convex and compact, and $Y$ convex, and $f$ is bounded, convex and continuous in the first argument, and concave in the first argument, $$\inf_{x \in X} \sup_{y \in Y} f(x,y) \leq \sup_{y \in Y} \inf_{x \in X} f(x,y).$$
then we can also show that
When $X$ and $Y$ are unit balls in an infinite dimensional Hilbert space, and $f$ is convex and 1-Lipschitz in the first argument, and concave in the second argument, $$\inf_{x \in X} \sup_{y \in Y} f(x,y) \leq \sup_{y \in Y} \inf_{x \in X} f(x,y).$$