Multilinear Map - k-Linear Map

• is a function of several variables that is linear separately in each variable
  • 𝑓: 𝑉₁ ⨯ ... ⨯ 𝑉_𝑛 → 𝑋
• a multilinear map of:
  • one variable is a linear map 𝑓: 𝑉₁ → 𝑋
  • two variables is a bilinear map 𝑓: 𝑉₁ ⨯ 𝑉₂ → 𝑋
  • k variables is a k-linear map 𝑓: 𝑉₁ ⨯ ... ⨯ 𝑉_𝑘 → 𝑋
• a multilinear map of:
  • k copies of the same variable and with a scalar field 𝐹 as its codomain is a multilinear form 𝑓: 𝑉 ⨯ ... ⨯ 𝑉 → 𝐹

Multilinear Map - Definition

A multilinear map is a function:

• 𝑓: 𝑉₁ ⨯ ... ⨯ 𝑉_𝑛 → 𝑋

such that:

• 𝑉₁, ..., 𝑉_𝑛, and 𝑋 are vector spaces
• for each 𝑖, if all the variables but 𝑣_𝑖 are held constant then 𝑓(𝑣₁, ..., 𝑣_𝑖, ..., 𝑣_𝑛) is a linear function of 𝑉_𝑖 to 𝑋

Resources

• https://en.wikipedia.org/wiki/Multilinear_map