Compact operators send bounded sets to precompact sets; on Hilbert space they are norm-limits of finite-rank operators.Suggest a slogan
Let \(X\) and \(Y\) be normed vector spaces. A linear map \(T\colon X\to Y\) is compact if \(T(B)\) has compact closure in \(Y\) for every bounded subset \(B\subseteq X\).
Every compact linear map \(T\colon X\to Y\) between normed vector spaces is bounded.
The closure of \(T(\theset{x\in X\suchthat\norm{x}\leq 1})\) is compact, hence bounded, so \(\sup_{\norm{x}\leq 1}\norm{Tx}<\infty\).
Let \(H\) be a Hilbert space. The set of compact operators \(H\to H\) is the closure, in the operator norm, of the set of bounded operators \(H\to H\) of finite rank.
What is a compact operator?