Definition of finite dimensional representation of a compact group

Question

Consider the following fragment:

One asks that the map $\pi: G \to GL(V)$ is continuous. What does this mean? If $V$ is finite-dimensional, then we can view $GL(V)$ as matrices and then I guess it means that all the matrix entries are continuous?

What does continuity mean when $V$ is a Banach space (not necessarily finite-dimensional)? Which topology is put on $GL(V)$? Is it pointwise convergence topology?

Answer 1

One of the most important representations of the additive group $\mathbb R$ takes place in the Hilbert space $L^2(\mathbb R)$ and is given by $$ \pi _s(\xi )|_t = \xi (t-s), \quad \forall \xi \in L^2(\mathbb R),\ \forall t,s\in \mathbb R. $$ This representation is not continuous with the norm topology on operators, but it is continuous with the strong (i.e. pointwise) topology. For that reason, most results about representations of topological groups assume strong continuity!

For a general locally compact group $G$, there exists a left invariant measure $\mu $ on $G$ (called the Haar measure), so we may consider a similar representation, this time of $G$ on $L^2(G, \mu )$, given by $$ \pi _s(\xi )|_t = \xi (s^{-1}t), \quad \forall \xi \in L^2(G, \mu ),\ \forall t,s\in G. $$ Unless $G$ is discrete, this representation is not norm-continuous but again it is strongly continuous.

Answer 2

To actually answer the question "What does continuity mean when $V$ is a Banach space" we need something stronger, that your text does not seem to mention:

We require that the group action of $G$ on $V$ defined by

$$G \times V \to V: (g, v) \mapsto \pi(g)v$$ is continuous. Here, $G \times V$ has the product topology.