Vector space/Tensor product/Generating system/Basis/Fact/Proof

Proof

(1). Due to the construction, the decomposable tensors ${}v_{1}\otimes _{}\cdots \otimes _{}v_{n}$ form a generating system of the tensor product. Hence, it is enough to show that they are linear combinations of the given family. But this follows from fact  (3).

(2). We can restrict to finite families. We want to apply fact. Let ${}(r_{1},\ldots ,r_{n})\in J_{1}\times \cdots \times J_{n}$ be fixed. Because of the linear independence of the families ${}v_{ij}$, ${}j\in J_{i}$, in ${}V_{i}$, there exist linear forms

$\varphi _{i}\colon V_{i}\longrightarrow K$

with ${}\varphi _{i}(v_{ir_{i}})\neq 0$ and with ${}\varphi _{i}(v_{ij})=0$ for ${}j\neq r_{i}$. Therefore,

$V_{1}\times \cdots \times V_{n}\longrightarrow K,(w_{1},\ldots ,w_{n})\longmapsto \varphi _{1}(w_{1})\cdots \varphi _{n}(w_{n}),$

is, according to exercise, a multilinear mapping. Due to fact, we have a corresponding linear mapping

$V_{1}\otimes _{}\cdots \otimes _{}V_{n}\longrightarrow K,$

which sends ${}v_{1r_{1}}\otimes _{}\cdots \otimes _{}v_{nr_{n}}$ to

${}\varphi _{1}(v_{1r_{1}})\cdots \varphi _{n}(v_{nr_{n}})\neq 0\,,$

and all other elements ${}v_{1j_{1}}\otimes _{}\cdots \otimes _{}v_{nj_{n}}$ of the family to ${}0$.

(3) follows from (1) and (2).

To fact