$G$-invariant complement to an infinite dimensional vector space Let $G$ be a finite group and let 
$$\rho : G \to GL(V)$$
be a complex representation of $G$. 
Suppose we have an internal direct sum decomposition 
$$V=W \oplus U$$
where $W$ is infinite dimensional and $U$ is finite dimensional and nonzero. Assume further that $W$ is $G$-invariant. 
Does there exist a subspace $U' \subseteq V$ such that $U'$ is $G$-invariant and $V=W \oplus U'$?
 A: Yes, there is such a $U'$, by Maschke's Theorem, which states that if $V$ is a representation of a finite group over a field of characteristic zero, and $W$ is a subrepresentation, then it has a complement $U'$ that is also a subrepresentation. Some introductions to representation theory might only state Maschke's Theorem for finite dimensional representations, as they will often only be concerned with the finite dimensional case, but if you look at the usual proof then you'll see that it nowhere uses the fact that $W$ is finite dimensional.
Also, there is no need to assume that $W$ has a finite dimensional vector space complement $U$.
A: The answer is no, not in general.  Of course this is not true in the finite dimensional case, and making $W$ infinite dimensional does nothing to help the situation. Here's kind of a contrived example:
Let's say $G$ acts on trivially on the infinite dimensional vector space $W$.
Now let $K$ be a finite dimensional representation of $G$ with a $G$-invariant subspace $U$, but such that $U$ does not have a $G$-invariant complement.
Now $K\oplus W$ is an infinite dimensional representation of $G$ with an infinite dimensional $G$ invariant subspace $U\oplus W$, but this subspace has no $G$-invariant complement.
