
Appendix F
Hilbert Spaces and Linear Operators
John von Neumann gave the first precise formulation of quantum mechanics in 1928/1929
– and a systematic account in his famous book [1119] from 1932 – using the formalism of
unbounded self-adjoint operators in Hilbert space. A modern presentation can be found in
[926,1059,1131]. In this appendix we briefly recall some of the basic mathematics, following
closely [926, 1059, 1131].
F.1 Basic Facts on Hilbert Spaces
To start with, let us recall that an inner product space is a complex vector space V in
which a complex valued function (·, ·):V × V → C is defined which satisfies
(i)(x, x) ≥ 0and(x, x) = 0 if and