Discontinuous linear map and AC

The question arises when I am constructing an elementary proof for the following claim:

Given a normed vector space $V$, the following are equivalent:

1. $V$ is finite dimensional
2. Every linear map $T:V\to V$ is continuous

Now, the direction $(1)\Rightarrow(2)$ is obvious, but when trying to show $(2)\Rightarrow(1)$, I notice that a part of my proof says "fix a linearly independent set $\{v_i:i\in\mathbb{N}\}$"; some obvious consideration shows that this is a weak form of choice, since the construction would fail, for example, in a universe with an infinite set which is Dedekind-finite. So what would be the exact form of choice equivalent to this statement, assuming ZF?