Reference for a vector space lemma of Hopf? I've been told that the following is due to Hopf. Let $A, B, C$ be complex vector spaces. Given any linear map 
$$v:A\otimes B \rightarrow C,$$
where $A, B, C$ are complex vector spaces and $v$ is injective when restricted to $a \otimes B$ or $A\otimes b$ for any fixed $a$ or $b$, then 
$$\dim v(A\otimes B)\ge \dim A + \dim B -1.$$
The cool part is that this is false for real vector spaces. The proof is supposedly topological in nature.
I could not manage to locate a reference for this theorem using Google. Is it in any standard text? I am looking for a proof and historical context.
 A: This is a duplicate of a question on Mathoverflow.
References, including some containing a proof, may be found there. It appears Hopf did not prove this, after all. In order to have it more widely available, let me record the argument given in the paper by Larry Smith linked in the MathOverflow question.
The given map is equivalent to a bilinear form. The non-degeneracy induces a map
$$S^{2a-1}\times S^{2b-1}\rightarrow S^{2c-1}.$$
Let $S^1\times S^1$ act on the domain by multiplication in each component. Then the multiplication map $S^1\times S^1\rightarrow S^1$ induces a map
$$\mathbb C P^{a-1}\times \mathbb C P^{b-1} \rightarrow \mathbb C P^{c-1}.$$
Restricting, this gives a map
$$\mathbb C P^{a-1}\vee \mathbb C P^{b-1} \rightarrow \mathbb C P^{c-1}$$
and this is homotopic to "the canonical inclusion on each factor of the wedge." Then we get, if $u,v,w$ are the generators of the cohomology rings of these spaces,
$$f^*(w)=u+v,$$
and $f^*(w^{a+b-2})$ is, up to a binomial coefficient, $u^{a-1}v^{b-1}$, which is nonzero. So $a+b-2\le c-1$, as desired.
[Feel free to edit this answer to clean up the proof.]
A: I think I have found a counterexample for $A = B = C = \mathbb{R}^2$.
We define $v$ via
$$v(a \otimes b) = (a^\top I \, b, a^\top R \, b),$$
where $I$ is the identity and
$$R = \begin{pmatrix}0 & 1 \\ -1 & 0 \end{pmatrix}$$
is a rotation by $\pi/2$. The injectivity on the factors follows, since $I \, b$ and $R \, b$ are linearly independent for $b \ne 0$ and span $\mathbb{R}^2$. Similarly, $I \, a$ and $R^\top a$ are linearly independent for $a \ne 0$.
But the same construction should also work for $\mathbb{C}^2$ (at least, I do not see any obstruction). Hence, I suspect that there is some error.
