Editing Multilinear map (section)

==Example==
Let's take a trilinear function

:<math>g\colon R^2 \times R^2 \times R^2 \to R, </math>
where {{math|1=''V<sub>i</sub>'' = ''R''<sup>2</sup>, ''d<sub>i</sub>'' = 2, ''i'' = 1,2,3}}, and {{math|1=''W'' = ''R'', ''d'' = 1}}.

A basis for each {{mvar|V<sub>i</sub>}} is <math>\{\textbf{e}_{i1},\ldots,\textbf{e}_{id_i}\} = \{\textbf{e}_{1}, \textbf{e}_{2}\} = \{(1,0), (0,1)\}.</math> Let

:<math>g(\textbf{e}_{1i},\textbf{e}_{2j},\textbf{e}_{3k}) = f(\textbf{e}_{i},\textbf{e}_{j},\textbf{e}_{k}) = A_{ijk},</math>
where <math>i,j,k \in \{1,2\}</math>. In other words, the constant <math>A_{i j k}</math> is a function value at one of the eight possible triples of basis vectors (since there are two choices for each of the three <math>V_i</math>), namely: 
:<math>
\{\textbf{e}_1, \textbf{e}_1, \textbf{e}_1\}, 
\{\textbf{e}_1, \textbf{e}_1, \textbf{e}_2\}, 
\{\textbf{e}_1, \textbf{e}_2, \textbf{e}_1\},
\{\textbf{e}_1, \textbf{e}_2, \textbf{e}_2\},
\{\textbf{e}_2, \textbf{e}_1, \textbf{e}_1\}, 
\{\textbf{e}_2, \textbf{e}_1, \textbf{e}_2\}, 
\{\textbf{e}_2, \textbf{e}_2, \textbf{e}_1\},
\{\textbf{e}_2, \textbf{e}_2, \textbf{e}_2\}.
</math>

Each vector <math>\textbf{v}_i \in V_i = R^2</math> can be expressed as a linear combination of the basis vectors

:<math>\textbf{v}_i = \sum_{j=1}^{2} v_{ij} \textbf{e}_{ij} = v_{i1} \times \textbf{e}_1 + v_{i2} \times \textbf{e}_2 = v_{i1} \times (1, 0) + v_{i2} \times (0, 1).</math>

The function value at an arbitrary collection of three vectors <math>\textbf{v}_i \in R^2</math> can be expressed as
:<math>g(\textbf{v}_1,\textbf{v}_2, \textbf{v}_3) = \sum_{i=1}^{2} \sum_{j=1}^{2} \sum_{k=1}^{2} A_{i j k} v_{1i} v_{2j} v_{3k},</math>
or in expanded form as
:<math> \begin{align}
g((a,b),(c,d)&, (e,f)) = ace \times g(\textbf{e}_1, \textbf{e}_1, \textbf{e}_1) + acf \times g(\textbf{e}_1, \textbf{e}_1, \textbf{e}_2) \\
&+ ade \times g(\textbf{e}_1, \textbf{e}_2, \textbf{e}_1) +
adf \times g(\textbf{e}_1, \textbf{e}_2, \textbf{e}_2) +
bce \times g(\textbf{e}_2, \textbf{e}_1, \textbf{e}_1) +
bcf \times g(\textbf{e}_2, \textbf{e}_1, \textbf{e}_2) \\ 
&+ bde \times g(\textbf{e}_2, \textbf{e}_2, \textbf{e}_1) +
bdf \times g(\textbf{e}_2, \textbf{e}_2, \textbf{e}_2).
\end{align} 
</math>