Characteristic property of the tensor product/Statement

Notice: this page is supposed to be transcluded, use full=true to show claims and extra things

Statement

$\xymatrix{ V_1\times\cdots\times V_k \ar@2{->}[rr]^-A \ar@2{->}[d]_p & & W \\ V_1\otimes\cdots\otimes V_k \ar@{.>}[urr]_-{\overline{A} } }$
Diagram of the situation, the double-arrows is multilinear, the other is linear

Let $\mathbb{F}$ be a field and let $\big((V_i,\mathbb{F})\big)_{i\eq 1}^k$ be a family of finite dimensional vector spaces over $\mathbb{F}$. Let $(W,\mathbb{F})$ be another vector space over $\mathbb{F}$. Then:

• If $A:V_1\times\cdots\times V_k\rightarrow W$ be any multilinear map
  • there exists a unique linear map, $\overline{A}:V_1\otimes\cdots\otimes V_k\rightarrow X$ such that:
    • $\overline{A}\circ p\eq A$ (that is: the diagram on the right commutes)

Where $p:V_1\times\cdots\times V_k\rightarrow V_1\otimes\cdots\otimes V_k$ by $p:(v_1,\ldots,v_k)\mapsto v_1\otimes\cdots\otimes v_k$ (and is $p$ is multilinear)