I am trying to understand the first part of Lemma 1.5.2 from Werner Greub's Multilinear Algebra. The lemma is stated for vector spaces $E,F,T$ and a bilinear map $\otimes$ satisfying the universal property with $\otimes: E \times F \to T$.
Lemma 1.5.2: Let $\{e_\alpha\}_{\alpha\in A}$ be a basis of E. Then every vector $z \in T$ can be written $z = \sum_\alpha e_\alpha \otimes b_\alpha$ with $b_\alpha \in F$ where only finitely many $b_\alpha$ are nonzero. Moreover, all $b_\alpha$ are determined uniquely by $z$.
Grueb gives two criteria for a bilinear map $\otimes$ to satisfy the universal property:
$\otimes_1$: The vectors $x\otimes y$ with $x\in E,y\in Y$ generate $T$.
$\otimes_2$: For any bilinear map $\varphi:E\times F\to H$ to any vector space $H$ there exists a linear map $f:T \to H$ such that $f\circ\otimes = \varphi$.
I do not understand why only finitely many $b_\alpha$ are nonzero. Grueb invokes $\otimes_1$ and claims we can write any $z=\sum_{v=1}^r x_v \otimes y_v$ where $x_v \in E,y_v\in F$. But how does the first part of the universal property alone ensure this is a finite sum?
The word "generate" just means everything can be written as a FINITE linear combination of the generators.
So the property $\otimes_1$ says that every vector $z$ in $T$ is a finite linear combination of vectors of the form $x\otimes y$, which he then writes as $\sum_{v=1}^r x_v \otimes y_v$.
By the way, usually people combine these two properties by requiring the map $f$ in $\otimes_2$ to be unique.