representations of compact groups are equivalent to unitary representations

Let $G$ be a compact topological group. If $(\pi,V)$ is a finite-dimensional representation (http://planetmath.org/TopologicalGroupRepresentation) of $G$ in a normed vector space $V$, then $\pi$ is equivalent (http://planetmath.org/TopologicalGroupRepresentation) to a unitary representation.

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Proof: Let $\langle\cdot,\cdot\rangle$ denote an inner product in $V$. Define a new inner product in the vector space $V$ by

 $\displaystyle\langle v,w\rangle_{1}:=\int_{G}\big{\langle}\pi(s)v,\pi(s)w\big{% \rangle}\;d\mu(s)$

where $\mu$ is a Haar measure in $G$. It is easy to see that $\langle\cdot,\cdot\rangle_{1}$ defines indeed an inner product, noting that $\langle\pi(\cdot)v,\pi(\cdot)w\rangle$ is a continuous function in $G$.

Now we claim that, for every $s\in G$, $\pi(s)$ is a unitary operator for this new inner product. This is true since

 $\displaystyle\big{\langle}\pi(t)v,\pi(t)w\big{\rangle}_{1}$ $\displaystyle=$ $\displaystyle\int_{G}\big{\langle}\pi(s)\pi(t)v,\pi(s)\pi(t)w\big{\rangle}\;d% \mu(s)$ $\displaystyle=$ $\displaystyle\int_{G}\langle\pi(st)v,\pi(st)w\rangle\;d\mu(s)$ $\displaystyle=$ $\displaystyle\int_{G}\langle\pi(s)v,\pi(s)w\rangle\;d\mu(s)$ $\displaystyle=$ $\displaystyle\langle v,w\rangle_{1}$

Denote by $V^{\prime}$ the space $V$ endowed the inner product $\langle\cdot,\cdot\rangle_{1}$. As we have seen, $(\pi,V^{\prime})$ is a unitary representation of $G$ in $V^{\prime}$. Of course, $(\pi,V^{\prime})$ and $(\pi,V)$ are equivalent representations, since

 $\displaystyle\pi=\mathrm{id}^{-1}\,\pi\,\mathrm{id}$

where $\mathrm{id}:V\longrightarrow V^{\prime}$ is the identity mapping. Thus, $(\pi,V)$ is equivalent to a unitary representation. $\square$

Title representations of compact groups are equivalent to unitary representations RepresentationsOfCompactGroupsAreEquivalentToUnitaryRepresentations 2013-03-22 18:02:28 2013-03-22 18:02:28 asteroid (17536) asteroid (17536) 9 asteroid (17536) Theorem msc 43A65 msc 22A25 msc 22C05