I am self-studying functional analysis and have just learned the definition of compact operators. However, it isn't clear to me why the name "compact operator" was chosen.
The operator itself could be considered a one-point subset of a topological vector space of operators, but the "compactness" doesn't seem to refer to any such topology (every operator would constitute a compact subset of such a space, I think).
The definition involves compactness, but only in an indirect way (bounded sets get mapped to relatively compact subsets). Thus it is unclear to me if there is a good reason for the term.
Well, as my professor says, compact operators are small and Fredholm operators are big since cokernel of a Fredholm operator should be finite dimensional while the image of a compact operator between Banach spaces can not contain an infinite dimensional closed subspace.
There is also a deeper connection between two of those types of operators which is a subject of Riesz--Schauder theory.