It seems that functional analysis is greatly motivated by quantum mechanics, right? Is there any reference on the relation between the two?
$\begingroup$ Quantum Turing Machines were inspired by quantum mechanics, as well quantum-resistant cryptography. $\endgroup$– PyRulezNov 2, 2017 at 19:26
The main contribution of quantum mechanics to functional analysis is the spectral theory of unbounded operators developed by John von Neumann (with a little help from Erhard Schmidt) and Marshall Stone. See this article by Kreyszig and Birkhoff.
Read Jean Dieudonné's History of Functional Analysis. More precisely, read §VII.4: The spectral theory of von Neumann. Chapter I of von Neumann's Mathematical foundations of Quantum Mechanics is very interesting too.
I also recommend Dieudonne (following Jose Carlos Santos) for a brief historical account, and also von Neumann's own classic book Mathematical foundations of quantum mechanics. And volumes by Reed and Simon, Methods of Mathematical Physics., if you want some later development. The influence was (and is) very strong, in particular the theory of unbounded operators was developed mostly to satisfy the demands of quantum mechanics.