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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?

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    $\begingroup$ Operator theory studies linear operators acting on vector spaces of functions (equipped with some kind of topology to make sense of infinite sums). Such vector spaces are a key object of study in functional analysis generally, but you can also study nonlinear functions (or "functionals") on them, rather than linear transformations; an example is the calculus of variations. $\endgroup$ Commented Jun 12, 2016 at 6:54
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    $\begingroup$ Note that there is a difference between "function space" and "functional space." In fact, a "functional" usually means a (possibly nonlinear) real or complex valued function defined on a function space. $\endgroup$ Commented Jun 12, 2016 at 6:59
  • $\begingroup$ Note that despite functionals act on functions, they are not that complicated as functors from lambda calculus. Dual space is the space of functionals acting on the original space of functions in a beautiful linear way. $\endgroup$ Commented Aug 20, 2020 at 21:38

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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.

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