I've read many questions about the fact that a closed and bounded ball in a generic Hilbert space is not compact. But no one furnishing a concrete example of a compact subset that can be used in applications. So I'm asking for some examples of compact sets in an infinite dimensional Hilbert (or only Banach) space.
In particular, given a bounded set $E$ in the Hilbert space $H$, does exist a compact $K$ such that $E\subset K$? Can we construct an exhaustion by compact sets for any Hilbert space?