I was wondering what is the meaning of a group acting linearly on a topological space like "circle acting linearly on the $k$-disk $D^k$". I know the meaning of a linear group action on a vector space but for arbitrary topological space, I don't know. Some interesting example will be helpful.
Thank you in advance...
The most natural definition is: An action $\times X\to X$ is linear (or, more precisely, is linearizable), if there exists an embedding $f: X\to V$ (where $V$ is a topological vector space) and a continuous linear action $G\times V\to V$ such that
$$ f(gx)= g f(x) $$ for all $x\in X$. Here it is negotiable what kind of a topological vector space you may want to consider, say, finite dimensional (in which case continuity is automatic) or Banach, or Frechet,... One may also want to impose further restrictions on the (topological) dimension of $X$ and the dimenson of $V$...
In other words, you can think of $X$ as "sitting inside" $V$ such that the action of $G$ on $X$ is the restriction of a (continuous) linear action of $G$ on $V$.