I am a little bit uncertain about an argumentation showing that a given map of a topological group is somehow obviously continuous. In the following I will rely on the book of Anthony W. Knapp „Lie Groups, Lie Algebras and Cohomology“.
Let us fix some notation and definitions;
Let $G$ be a topological group. A finite-dimensional representation of $G$ is a homomorphism $\pi$ of $G$ into the group of invertible linear maps of a finite-dimensional complex vector space $V$ into itself such that the resulting map of $G \times V$ into $V$ is continuous.
Now I would like to understand the following assertion:
Let $G$ be topological group, and let $\sigma$ and $\rho$ be representations of $G$ on finite-dimensional complex vector spaces $V$ and $W$, respectively. Define a representation $\pi$ of $G$ on $V \otimes_{\mathbb{C}} W$ by
$$\pi(g) = \sigma(g) \otimes \rho(g).$$
Then one can use bases to see that $\pi$ is continuous.
I don't understand the last sentence; how it is supposed to help me to show continuity of the resulting map $G \times (V \otimes_{\mathbb{C}} W) \to V \otimes_{\mathbb{C}} W$. For fixed $g \in G$ one can argue that $\pi(g)$ is a linear map between finite-dimensional vector spaces and therefore continuous. But what about the parameter $g$?
I have to show that the map $(g,m \otimes n) \mapsto \sigma(g) \otimes \rho(g)(m \otimes n) = \sigma(g)(m) \otimes \rho(g)(n)$ is continuous in $(g,m \otimes n) \in G \times (V \otimes_{\mathbb{C}} W)$ and I am therefore obliged to write it as a composition of continuous maps. I know that $\sigma$ and $\rho$ are continuous but I don't know nothing about the continuity of $\otimes$ and that is where I fail.
I appreciate any sort of suggestion which helps me to understand this somewhat trivial argumentation.
Best Answer
Lemma. If $V$ is a complex representation of the underlying group of $G$, then $G \times V \to V$, $(g,v) \mapsto gv$ is continuous if and only if the homomorphism $G \to \mathrm{GL}(V)$, $g \mapsto (v \mapsto gv)$ is continuous.
Once you have proven this general and useful Lemma, it is easy to construct tensor products. Namely, if $G$ acts on $V$ and on $W$, then we obtain an action of $G$ on $V \otimes W$ via $$G \to G \times G \to \mathrm{GL}(V) \times \mathrm{GL}(W) \to \mathrm{GL}(V \otimes W).$$ Thus, we only have to prove that $\mathrm{GL}(V) \times \mathrm{GL}(W) \to \mathrm{GL}(V \otimes W)$ is continuous. But this is even a polynomial map, when you write it down in bases.