Tensor product of vector spaces

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Currently in the notes stage, see Notes:Tensor product

Any first-time readers should look at the abstract definition first

Definition

Let $\mathbb{F}$ be a field and let $\big((V_i,\mathbb{F})\big)_{i\eq 1}^k$ be a family of vector spaces over $\mathbb{F}$. Let $\mathcal{F}(V_1\times\cdots\times V_k)$ denote the free vector space on $\prod_{i\eq 1}^kV_k$. We define the (abstract) tensor product of $V_1,\ldots,V_k$ as:

• $V_1\otimes\cdots\otimes V_k:\eq\dfrac{\mathcal{F}(V_1\times\cdots\times V_k)}{\mathcal{R} }$ where $\mathcal{R}$ is defined as follows:
  • $\mathcal{R}$ denotes the span of all the union of the following two sets:
    1. $\big\{ (v_1,\ldots,v_{i-1},av_i,v_{i+1},\ldots,v_k)-a(v_1,\ldots,v_k)\ \big\vert\ i\in\{1,\ldots,k\}\wedge a\in\mathbb{F}\wedge\forall j\in\{1,\ldots,k\}[v_j\in V_j]\big\}$
    2. $\big\{(v_1,\ldots,v_{i-1},v_i+v'_i,v_{i+1},\ldots,v_k)-(v_1,\ldots,v_k)-(v_1,\ldots,v_{i-1},v'_i,v_{i+1},\ldots,v_k)\ \big\vert\ i\in\{1,\ldots,k\}\wedge v'_i\in V_i\wedge\forall j\in\{1,\ldots,k\}[v_j\in V_j]\big\}$

Basis

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. Let $n_i:\eq\text{Dim}(V_i)$ and $e^{(i)}_1,\ldots,e^{(i)}_{n_i}$ denote a basis for $V_i$, then we claim:

• $$\mathcal{B}:\eq\left\{e^{(1)}_{i_1}\otimes\cdots\otimes e^{(k)}_{i_k}\ \big\vert\ \forall j\in\{1,\ldots,k\}\subset\mathbb{N}[1\le i_j\le n_j]\right\}$$

Is a basis for the tensor product of the family of vector spaces, $V_1\otimes\cdots\otimes V_k$

Note that the number of elements of $\mathcal{B}$, denoted $\vert\mathcal{B}\vert$, is $\prod_{i\eq 1}^kn_i$ or $\prod_{i\eq 1}^k\text{Dim}(V_i)$, thus:

• $\text{Dim}(V_1\otimes\cdots\otimes V_k)\eq\prod_{i\eq 1}^k n_i$

Characteristic property

$\xymatrix{ V_1\times\cdots\times V_k \ar@2{->}[rr]^-A \ar@2{->}[d]_p & & W \\ V_1\otimes\cdots\otimes V_k \ar@{.>}[urr]_-{\overline{A} } }$
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 $A:V_1\times\cdots\times V_k\rightarrow W$ is 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)