Multilinear map

This is a generalisation of linear maps and bilinear maps

Definition

For a [ilmath]k\in\mathbb{N} [/ilmath] a [ilmath]k[/ilmath]-linear map is a mapping, [ilmath]f[/ilmath] from [ilmath]k[/ilmath] vector spaces [ilmath]U_1,\cdots,U_k[/ilmath] all over a field [ilmath]F[/ilmath] to another vector space [ilmath](V,F)[/ilmath]:

• [ilmath]f:U_1\times\cdots\times U_k\rightarrow V[/ilmath]

Where, for every [ilmath]i\in\{1,\cdots,k\}[/ilmath], [ilmath]x_i,y_i\in U_i[/ilmath] and [ilmath]\lambda,\mu\in F[/ilmath] we have:

• [ilmath]f(x_1,\cdots,x_{i-1},\lambda x_i+\mu y_i,x_{i+1},\cdots,x_k)=\lambda f(x_1,\cdots,x_{i-1},x_i,x_{i+1},\cdots,x_k)+\mu f(x_1,\cdots,x_{i-1},x_i,x_{i+1},\cdots,x_k)[/ilmath]

So basically it's just linear in each term.

See next

• Space of all [ilmath]k[/ilmath]-linear mappings - [ilmath]L(U_1,\cdots,U_k;V)[/ilmath] - which is a vector space

See also

• Tensor product
• Linear map
• Bilinear map