Compact operators
The spectral theory and its applications. Generalities and compact operators

By definition, a compact operator transforms the unit ball into a relatively compact set, i.e. one with compact adherence; by virtue of the Bolzano-Weierstraβ theorem, this again means that from the image of any bounded sequence, we can extract a convergent sub-sequence. The specific properties of compact operators are based on the following result: a Banach space X is of finite dimension if and only if its unit ball is compact, or X is locally compact. The proximity between compact operators and operators acting on finite-dimensional spaces is therefore not surprising.

Note, that a compact operator S transforms a weakly convergent sequence into a strongly convergent one, indeed according to the Banach-Steinhaus theorem, a weakly convergent...