In this article, there is a lemma as following:
Let $U$ and $V$ be vector spaces, and let $b:U\times V\to X$ be a bilinear map from $U\times V$ to a vector space $X$. Suppose that for every bilinear map $f$ defined on $U\times V$ there is a unique linear map $c$ defined on $X$ such that $f=cb$. Then there is an isomorphism $i:X\to U\otimes V$ such that $u\otimes v=ib(u,v)$ for every $(u,v)$ in $U\otimes V$.
There are several other places in this article where the author uses "every bilinear map $f$ defined on $U\times V$ " without specifying the range of the function.
As I understand, it may mean one of the following items:
- $\forall f\in{\mathcal L}(V,W;Y)$ for some vector space Y;
- $\forall f\in \bigcup_{Y\text{is a vector space}}{\mathcal L}(V,W;Y).$
Which one is correct and how should I understand it?