View source for Basis for the tensor product/Statement

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==Statement==
</noinclude>Let {{M|\mathbb{F} }} be a [[field]] and let {{M|\big((V_i,\mathbb{F})\big)_{i\eq 1}^k}} be a family of ''[[dimension (vector space)|finite dimensional]]'' [[vector spaces]]. Let {{M|n_i:\eq\text{Dim}(V_i)}} and {{M|e^{(i)}_1,\ldots,e^{(i)}_{n_i} }} denote a [[basis (vector space)|basis]] for {{M|V_i}}, then we claim{{rITSMJML}}:
* {{MM|\mathcal{B}:\eq\left\{e^{(1)}_{i_1}\otimes\cdots\otimes e^{(k)}_{i_k}\ \big\vert\ \forall j\in\{1,\ldots,k\}\subset\mathbb{N}[1\le i_j\le n_j]\right\} }}
Is a {{link|basis|vector space}} for the [[tensor product]] of the family of vector spaces, {{M|V_1\otimes\cdots\otimes V_k}}


Note that the number of elements of {{M|\mathcal{B} }}, denoted {{M|\vert\mathcal{B}\vert}}, is {{M|\prod_{i\eq 1}^kn_i}} or {{M|\prod_{i\eq 1}^k\text{Dim}(V_i)}}, thus:
* {{M|\text{Dim}(V_1\otimes\cdots\otimes V_k)\eq\prod_{i\eq 1}^k n_i}}{{rITSMJML}}<noinclude>
==References==
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