Multilinear map

This is a generalisation of linear maps and bilinear maps

Definition

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

• $f:U_1\times\cdots\times U_k\rightarrow V$

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

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

So basically it's just linear in each term.

See next

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

See also

• Tensor product
• Linear map
• Bilinear map