# What's the motivation for the term "compact" operator?

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, the reason is precisely that bounded subsets map to relatively compact sets. There isn't any more reason to it. Commented Feb 5, 2021 at 21:30
• math.stackexchange.com/questions/3802255/… See the answer to this question. Commented Feb 6, 2021 at 0:12
• What's the motivation behind the term "bounded linear operator"? Bounded sets get mapped to bounded sets. I think "compact linear operator" just strengthens "bounded linear operator" by saying that bounded sets get mapped not only to merely bounded sets, but to (pre)compact sets. Commented Mar 4, 2021 at 22:51