View source for Tensor product of vector spaces

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: {{strike|Currently in the notes stage, see [[Notes:Tensor product]]}}
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: {{Note|Any first-time readers should look at the [[#Abstract definition|abstract definition]] first}}
==Definition==
Let {{M|\mathbb{F} }} be a [[field]] and let {{M|\big((V_i,\mathbb{F})\big)_{i\eq 1}^k}} be a family of [[vector spaces]] over {{M|\mathbb{F} }}. Let {{M|\mathcal{F}(V_1\times\cdots\times V_k)}} denote the [[free vector space generated by|free vector space]] on {{M|\prod_{i\eq 1}^kV_k}}. We define the (abstract) ''tensor product'' of {{M|V_1,\ldots,V_k}} as{{rITSMJML}}:
* {{M|V_1\otimes\cdots\otimes V_k:\eq\dfrac{\mathcal{F}(V_1\times\cdots\times V_k)}{\mathcal{R} } }}<!--

NOTE ABOUT THE SIZE OF F(V_1X...XV_k)

--><ref group="Note">Take a moment to respect just how vast the space {{M|\mathcal{F}(V_1\times\cdots\times V_k)}} is (especially if {{M|\mathbb{F}:\eq\mathbb{R} }} for example). Remember that this is ''not'' the space {{M|V_1\times\cdots\times V_k}} even though we write them as [[tuples]]. It is a ''huge'' space. 
* {{XXX|Flesh out this note}}</ref><!--

END OF SIZEOF F(V_1X...XV_k) NOTE

--> where {{M|\mathcal{R} }} is defined as follows:
** {{M|\mathcal{R} }} denotes the {{link|span|vector space}} of all the union of the following two [[sets]]:
**# {{M|\big\{ (v_1,\ldots,v_{i-1},av_i,v_{i+1},\ldots,v_k)-a(v_1,\ldots,v_k)\ \big\vert\ i\in\{1,\ldots,k\}\wedge a\in\mathbb{F}\wedge\forall j\in\{1,\ldots,k\}[v_j\in V_j]\big\} }}
**# {{M|\big\{(v_1,\ldots,v_{i-1},v_i+v'_i,v_{i+1},\ldots,v_k)-(v_1,\ldots,v_k)-(v_1,\ldots,v_{i-1},v'_i,v_{i+1},\ldots,v_k)\ \big\vert\ i\in\{1,\ldots,k\}\wedge v'_i\in V_i\wedge\forall j\in\{1,\ldots,k\}[v_j\in V_j]\big\} }}

==Abstract definition==
==[[Basis for the tensor product|Basis]]==
{{:Basis for the tensor product/Statement}}
==[[Characteristic property of the tensor product|Characteristic property]]==
{{:Characteristic property of the tensor product/Statement}}
==Notes==
<references group="Note"/>
==References==
<references/>
{{Definition|Linear Algebra|Abstract Algebra}}