Let $G$ be a compact subgroup of $O(n)$. Let $\rho$ be a continuous finite dimensional representation of $G$.



> 
> **Question** Is it true that there exists a continuous finite dimensional representation $\pi$ of $O(n)$ such that $\rho$ is a direct summand of the restriction of $\pi$ to $G$?
> 
> 
> 


I am almost sure that this is true. A reference would be very helpful.



Imbedding of a representation of a compact subgroup
I don't have a reference, but here is a short argument.
Assume $G<H$ are compact groups (in our example take $H=\text{O}(n)$)
and recall that unitary representations of compact groups are completely reducible. Recall also that by Peter-Weyl every irreducible unitary representation is finite dimensional.


Assume first that $\rho$ is an irreducible finite dimensional continuous representation of $G$ and endow it with a $G$-invariant inner product. Fix $\pi$ to be any finite dimensional subrepresentation of the unitary induction $\text{Ind}\_G^H(\rho)$, eg an irreducible subrepresentation.
Then, by Frobenius reciprocity, $\text{Hom}\_G(\rho,\text{Res}\_G^H \pi)\simeq \text{Hom}\_H(\text{Ind}\_G^H(\rho),\pi)$ and the latter is non-empty by our choice of $\pi$.
It follows that $\rho$ is a direct summand of $\text{Res}\_G^H \pi$ by its irreducibility.


The case of a general $\rho$, continuous finite dimensional representation, follows easily.


