If $A\otimes B\rightarrow C$ is a nondegenerate linear map, where $A, B, C$ are vector spaces over an algebraically closed field, then $\dim C\ge \dim A + \dim B -1$.

Nondegenerate here means that the map is injective on each factor, ie, if the image of $a\otimes b$ is zero then either $a$ or $b$ is zero.

This result is known (at least in the Algebraic Geometry community) as the "Hopf lemma" (see for instance this question). Arbarello-Cornalba-Griffits-Harris actually contextualize the statement (for $\mathbb C$) by saying "In fact, it is a remarkable theorem of H. Hopf that [theorem]. This result, which is false over $\mathbb R$, was one of the earliest applications of topology to algebra." (reference given by Yusuf Mustopa in the cited MO question).

After some unsuccessful searches, one encounters a few works of Hopf on "compositions of quadratic forms", notably

- H. Hopf,
*Ein topologischer Beitrag zur reellen Algebra*, Comment. Math. Helv. 13 (1940/41), 219–239.

The focus is on maps of real vector spaces, and the motivation seems to be the classification of division algebras over $\mathbb R$. I haven't been able to find the result for $\mathbb C$ there. On the other hand, Daniel Shapiro in the book

- D. B. Shapiro, Compositions of Quadratic Forms, W. de Gruyter Verlag, 2000.

(downloadable from the author's webpage) attributes the result for $\mathbb C$ to Larry Smith, and more precisely to the paper

- Smith, L.
*Nonsingular bilinear forms, generalized J homomorphisms, and the homotopy of spheres I*, Indiana Univ. Math. J. 27 (1978), 697–737.

Shapiro also includes a proof for an arbitrary algebraically closed field (Prop 14.25). Smith starts his proof (for $\mathbb C$) by saying "The following argument is inspired by the paper [9] of Hopf", ie, the 1940/41 paper.

So, did Hopf ever state the "lemma" above? If not, is it generally accepted that Smith's contribution was a minor modification, so justifying an attribution to Hopf? Or is there a reference older than Smith? The latter seems unlikely, as Shapiro's book seems very well documented. But at the same time, it is a bit strange that ACGH say "it was one of the earliest applications of topology to algebra" of a result proved in 1978.

On Petri's Analysis of the Linear System of Quadrics through a Canonical Curve(Math. Ann. 206, 1973). It is stated as "an easy lemma" (without proof). $\endgroup$