In the representation theory class, we proved that any unitary representation from group $G$ (assuming that $G$ is locally compact topological group equipped with Haar measure) acting on $GL(V)$ where $V$ is $n$ dimensional $\mathbb{C}-$vector space is semisimple (i.e. reducible into direct sum of irreducible subrepresentations). Moreover, we proved that any vector space with Hermitian form can admit unitary representation via Weyl's unitary trick.
Here, I am a bit confused with the concept of irreducibility. According to the argument above, reducibility of representation depends on what inner product (or more broadly, what structure) the vector space has. For instance, if we want to claim that vector space $V$ is direct sum of irreducible subrepresentations for any finite dimensional representation of group $G$, then shouldn't we need to specify what inner product we are equipping $V$ with so that the representation becomes unitary? Is this understanding correct, or am I missing something from the above?
Thanks!
A representation is irreducible if it contains no (proper, non-trivial) subrepresentations. There is no requirement for there to be an inner product. Indeed for non-compact groups we don't have to have any (finite dimensional) unitary representations at all.
In the compact case, you can discuss how the inner product interacts with the subrepresentations (i.e. you can decompose into a direct sum of orthogonal subrepresentations) but the decomposition isn't predicated on the existence of the inner product.