Let (𝑉𝑖)𝑛𝑖=1 and 𝑋 be vector spaces (or modules over a commutative ring).
A multilinear map𝑀 is a function 𝑀:∏𝑛𝑖=1𝑉𝑖→𝑋 such that it is linear in each argument 𝑉𝑖 when all others are held constant. #m/def/linalg
See also
Balanced product generalizes bilinearity to modules over non-commutative rings.