Is the matrix exponential independent of the norm?

The definition of the matrix exponential uses powers, multiplication by scalars, sums and a limit. The former 3 are given by the vector space the matrices are in. A limit would require making that space of matrices into a normed space, i believe. Since there are different possible norms for a space of matrices, it is not clear that the same sequences converge for each norm, which could mean that the matrix exponential depends on norm.

While writing this i remembered that a space of matrices is isomorphic (as a vector space) to $mathbb{F}^k$ for some $k in mathbb{N}$ and some field $mathbb{F}$ and all norms in $mathbb{F}^k$ are equivalent. I believe isomorphism as a vector space would imply that all norms are equivalent since norms are defined in terms of the vector space. Is this correct?

(Edit: My last claim might be wrong if what someone commented is true, but its at least true for the real and complex fields)