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.