Is $W$ a finite-dimensional vector space in Proposition 3.13?

Question

Is $W$ a finite-dimensional vector space in Proposition $3.13$?

Answer 1

The author seems to assume that $W$ is a vector space and also a smooth manifold. So yes, it is finite-dimensional.

Answer 2

Yes it is, and there is really no need to write down that explicitly. The theorem states that there is a canonical isomorphism $V\cong T_aV$ when $V$ is finite-dimensional, so in writing $W\cong T_{La}W$ it is implicitly assumed that $W$ is finite-dimensional as well as it is using this canonical isomorphism for $W.$