How do you define an operator set-theoretically?

Question

I'm not sure what operators are. Are they functions or relations? It is usually said that they are mappings, but, in my experience, "mappings" is an ambiguous phrase with some people using the word to refer to functions, others to relations and yet others to category-theoretic morphisms. I thought that a set-theoretic definition of an operator would clarify things for me. There are many types of operators, but I was wondering how operators in general are defined. Online resources didn't help me so far. Any resource recommendation on this would be very much appreciated.

Answer 1

The context is important. An operator is just a function, but if it takes functions to functions, then there is a possibility of confusion, and so another synonym is used. Sometimes they are called transforms instead. It is mostly based on long standing conventions. For example, the Laplace transform could have been called the Laplace operator, but isn't. Sometimes they are called functionals. There are other possibilities, but it doesn't change that an operator is really just a function of a certain type depending on context. For example, in another context, a unary operator (operation) is another name for an endofunction, a function from a set to itself, and a binary operator is another name for a function from the cartesian product of a set with itself to that set, and so on.