In page 82 of Brian C. Hall's Lie Groups, Lie Algebras and Representations, he establishes the following representation $\Pi_m$ of SU(2) on $V_m$, the space of homogeneous polynomials of degree $m$ in two complex variables:
For all $U\in SU(2),\;\Pi_m(U)\in GL(V_m)$ is given by its action on arbitrary $f\in V_m$:
$[\Pi_m(U)f](z) = f(U^{-1}z),\quad z\in \mathbb{C}^2$
Question:
It's quite simple to show this is indeed a group homomorphism. The book doesn't mention, however, about the continuity of this homomorphism (which is necessary since this is a representation of Lie groups not just a representation of groups). Can someone help me prove the continuity?
What I have done so far:
I tried to prove the continuity of $\Pi_m$ but haven't suceeded. I know everything is in terms of addition and multiplication in the formula, but then to say it's automatically continuous seems handwavy (or relying on some established result about continuity I don't know about).
For example, I see that $[\Pi_m(U)f]$ is continuous, since it's a composition of continuous functions. But my problem is that $\Pi_m$ is not into $V_m$, but into $GL(V_m)$. I tried for a bit to work on the fact that $GL(V_m)$ is isomorphic to $GL(\mathbb{C}^{m+1})$ and show the matrix entries of $\Pi_m(U)$ under this isomorphism are all continuous functions of the entries of $U\in SU(2)$. I have not succeeded but I think there should be an easier way.
If you're comfortable with dual vector spaces and tensor products, then it's not too difficult to make the following chain of associations, letting $V = \mathbb C^2$ be the vector space that $SU(2)$ canonically acts on, and noting that we don't really need to restrict to $SU(V)$; we can look at all of $GL(V)$: $$GL(V) \to GL(V^*) \to GL(T(V^*)) \rightsquigarrow GL(Sym(V^*)) \rightsquigarrow GL(Sym^m(V^*)),$$ where the first mapping is part of a contravariant functor, and the rest are covariant associations, and the squiggly arrows indicate that the linear automorphisms coming from earlier remain well-defined through the respective quotients, namely the quotient of algebras $Sym(V^*) = T(V^*)/(v \otimes w - w \otimes v)$ and the quotient of vector spaces $Sym(V^*) \twoheadrightarrow Sym^m(V^*)$ projecting onto the summand of the $m^\text{th}$ symmetric tensor power.
Each of these associations can be expressed in coordinates, and you'll readily see that they are therefore continuous. And then, in order to preserve the order of multiplication instead of reversing it, going back to $SU(2)$, we can take inverses.
The clue, by the way, that we operate with tensor powers of $V^*$ instead of with tensor powers of $V$, is that polynomials represent functions from $V$ to scalars.