What is the intuition/motivation behind compact linear operator.

Question

Compact Linear Operator is defined such that the operator will map any bounded set into a relatively compact set. Why is this property so special that it can be named as "compact"? Does it share some similar properties as compact sets? What is the motivation to define and study such a set?

Answer 1

The motivation to study compact operators is that a lot of the operators that we are interested in are compact.

The reason the property of being compact is so special, as you say, that it deserved to be given a name is that it has many useful and interesting consequences, of course.

In fact, you can generalize this: for almost all X and Y, the motivation to study objects of type X which have property Y is that lot of objects of type X that we are interested in have property Y.

Answer 2

The set of compact operators (in a Hilbert space) is exactly the set of norm limits of finite rank operators. This is perhaps a more natural definition than the one you indicate. Many of the nice properties of finite rank operators have analogues for compact operators. You can view compactness has a slight generalization of being finite rank that preserves (or only mildly weakens) most of these properties.

Above all, as Mariano notes, compact operators appear all the time "in the wild," so it is useful to develop their theory. You may be interested in the book History of Functional Analysis, which describes how functional-analytic abstractions like compactness arose from concrete physical problems and PDEs.