Facts about the tensor product

Question

I'm proving some statements about density operators, and would like to use two things. The problem is, that I'm not entirely sure they hold.

Let $V,\; W$ be finite dimensional, complex inner product spaces. The statements I want to use are:

(i) $\psi \in V\otimes W \; \Longrightarrow \; \exists \; v\in V, \; w\in W$ such that $\psi = v \otimes w.$

(ii) $\rho \in \text{End}(V\otimes W); \;\phi_1, \; \phi_2 \in \text{End}(V); \; \theta_1, \theta_2 \in \text{End} (W)$ such that $\psi = \phi_1 \otimes \theta_1 = \phi_2 \otimes \theta_2$ $ \Longrightarrow \; \phi_1 = \phi_ 2,\; \theta_1 = \theta_2$.

Does somebody know if these two are true?

Answer 1

Both are false as stated.

1. The set $\{ v \otimes w : v \in V, w \in W \}$ is not even a vector subspace of $V \otimes W$ in general. Try adding two elements together!
2. Consider the case where $V = 0$: then $V \otimes W = 0$ as well.