Can these characterisations of finite dimensionality be proven equivalent without using a basis?

Question

I was wondering about how to define "finite dimensional" without talking about bases. Two possibilities occurred to me:

1. Say $V$ is finite dimensional if the canonical inclusion $V\hookrightarrow V^{**}$ has an inverse.
2. Say $V$ is finite dimensional if the canonical inclusion $V\otimes V^*\hookrightarrow \text{End}(V)$ has an inverse.

It's easy enough to verify that these are indeed both equivalent to

3. Say $V$ is finite dimensional if it has a finite basis.

Question: Can we prove $(1)\Rightarrow (2)$ or $(2)\Rightarrow (1)$ without going via $(3)$?

Answer 1

Another possibility ( it uses topology ). A normed space is finite dimensional iff the unity ball is compact.