Editing Unitary representation (section)

==Formal definitions==

Let ''G'' be a topological group. A '''strongly continuous unitary representation''' of ''G'' on a Hilbert space ''H'' is a group homomorphism from ''G'' into the unitary group of ''H'',

:<math> \pi: G \rightarrow \operatorname{U}(H) </math>

such that ''g'' → π(''g'') ξ is a norm continuous function for every ξ ∈ ''H''.

Note that if G is a [[Lie group]], the Hilbert space also admits underlying smooth and analytic structures. A vector ξ in ''H'' is said to be '''smooth''' or '''analytic''' if the map ''g'' → π(''g'') ξ is smooth or analytic (in the norm or weak topologies on ''H'').<ref>
Warner (1972)</ref> Smooth vectors are dense in ''H'' by a classical argument of [[Lars Gårding]], since convolution by smooth functions of [[Support (mathematics)#compact support|compact support]] yields smooth vectors. Analytic vectors are dense by a classical argument of [[Edward Nelson]], amplified by Roe Goodman, since vectors in the image of a  heat operator ''e''<sup>–tD</sup>, corresponding to an [[elliptic differential operator]] ''D'' in the [[universal enveloping algebra]] of ''G'', are analytic. Not only do smooth or analytic vectors form dense subspaces; but they also form common cores for the unbounded skew-adjoint operators corresponding to the elements of the [[Lie algebra]], in the sense of [[spectral theory]].<ref>Reed and Simon (1975)</ref>

Two unitary representations π<sub>1</sub>: ''G'' → U(''H''<sub>1</sub>), π<sub>2</sub>: ''G'' → U(''H''<sub>2</sub>) are said to be '''unitarily equivalent''' if there is a [[unitary transformation]] ''A'':''H''<sub>1</sub> → ''H''<sub>2</sub> such that π<sub>1</sub>(''g'') = ''A''<sup>*</sup> ∘ π<sub>2</sub>(''g'') ∘ ''A'' for all ''g'' in ''G''. When this holds, ''A'' is said to be an [[intertwining operator]] for the representations <math>(\pi_1,H_1),(\pi_2,H_2)</math>.<ref>[[Paul Sally]] (2013) ''Fundamentals of Mathematical Analysis'', [[American Mathematical Society]] [https://books.google.com/books?id=b05c370fLdsC&pg=PA234 pg. 234]</ref>

If <math>\pi</math> is a representation of a connected Lie group <math>G</math> on a ''finite-dimensional'' Hilbert space <math>H</math>, then <math>\pi</math> is unitary if and only if the associated Lie algebra representation <math>d\pi:\mathfrak{g}\rightarrow\mathrm{End}(H)</math> maps into the space of skew-self-adjoint operators on <math>H</math>.<ref>{{harvnb|Hall|2015}} Proposition 4.8</ref>