In my undergrad mind they are the same subject because functional analysis studies functional spaces like Banach and Hilbert spaces. Operators are function, so shouldn't they be the same subject?
What is the distinction between the two fields?
I know many great books on functional analysis, such as one by Kreysig. What is a good introductory text on operator theory?
Best Answer
Actually I do not agree with some comments. In contemporary mathematics, operator theory is a branch of functional analysis that focuses on bounded and unbounded maps from a normed vector space (or a topological vector space) into another. Spectral theory is probably the most relevant part of operator theory, and it is linked to the theory of algebras ($C^*$-algebras, Banach algebras,...).
Functional analysis is the common name for many theories that share a common factor: the use of topological structures on vector spaces. So, the Hahn-Banach theorem is studied in a course of functional analysis, but it is seldom taught in a course of operator theory. The converse is usually false: spectral theory is often introduced in functional analysis, since operator theory is a more advanced course.
Many books introduce operator theory, like Conway's: just look up "operator theory" on Amazon or Google Books. If you want a book that presents functional analysis with a view towards operator theory, I can suggest the first (and the remaining three) books by Reed and Simon.