Let $K,T$ be fields and $V:=\{g:K\to T\}$ be a vector space over T. Then take $W:=V\otimes V$, is this $W$ isomorphic to some function space?

*Little background*: In quantum mechanics the the state of a one-electron (half spin fermion) system at a given time is a function $\mathbb{R}^{3}\to \mathbb{C}^4$ (with possible constraints I can't recall). Then the state of an $N$-electron system is an element of $\{\mathbb{R}^{3}\to \mathbb{C}^4\}^{\otimes N}$, then mysteriously the state is regarded as a function $\mathbb{R}^{3N}\to (\mathbb{C}^4)^{\otimes N}$.

Full disclosure: I posted this question on [Mathematics](https://math.stackexchange.com/q/1901217/351576) as well.