I often come across linear, continuous, bounded and/or compact operators, whose definition I am not very familiar with. I know that a linear operator is a map (a function) that preserves addition and scalar multiplication.
What is the relation between linear, continuous, bounded and compact operators? For example, is a continuous operator always linear or vice-versa?
I am looking for some guidelines: for example, a compact operator is always bounded. I would also appreciate if you provide (intuitive) explanations.
In functional analysis, the term operator almost always refers to a linear operator. Thus, operator and linear operator are synonyms.
It is an easy exercise to prove that linear operators are continuous iff bounded.
Some operators look very much like finite-dimensional ones; these are called compact. It happens that compact operators are always bounded.
Thus, we have this series of inclusions:
$$compact \subset bounded = continuous \subset linear$$
In case the domain is finite-dimensional, any operator is automatically continuous, bounded, and compact. Thus, the above hierarchy collapses to
$$compact = bounded = continuous = linear$$
This is the reason why one rarely talks about these definitions in finite dimensions (although the definitions themselves are still perfectly applicable).