# Multilinear Maps

Tags: maps, multilinear
 P: 530 Here is a bilinear map φ of two vector spaces V₁ & V₂ into another vector space N with respect to the bases B = {e₁,e₂} & B' = {e₁',e₂'}: φ : V₁ × V₂ → N | (m₁,m₂) ↦ φ(m₁,m₂) = φ(∑ᵢλᵢeᵢ,∑jμje'j) _______________________________= φ(λ₁e₁ + λ₂e₂,μ₁e₁' + μ₂e₂') = φ(λ₁e₁ + λ₂e₂,μ₁e₁') + φ(λ₁e₁ + λe₂,μ₂e₂') _______________________________= φ(λ₁e₁,μ₁e₁') + φ(λ₂e₂,μ₁e₁') + φ(λ₁e₁,μ₂e₂') + φ(λe₂,μ₂e₂') _______________________________= λ₁μ₁φ(e₁,e₁') + λ₂μ₁φ(e₂,e₁') + λ₁μ₂φ(e₁,e₂') + λ₂μ₂φ(e₂,e₂') _______________________________= ∑ᵢ∑jλᵢμjφ(eᵢ,ej') (Hope that's right!) Now in a discussion of bilinear (quadratic, hermitian) forms the terms φ(eᵢ,ej') in ∑ᵢ∑jλᵢμjφ(eᵢ,ej') are elements of a field, & there seems to be quite a lot of theory built up for the special case of N being a field, but thus far I can't really find any discussion of what happens when the φ(eᵢ,ej') elements are elements of a general vector space. Furthermore, the general case of multilinear (n-linear) maps expressed in this way seem to follow the same pattern of focusing on fields (or rings), e.g. determinants. The limited discussions of tensors & exterior algebra I've seen also follow this format of focusing on maps into a field. Just wondering what there is on maps of the form φ : V₁ × V₂ × ... × Vn→ N φ : V₁ × V₂ × ... × Vn→ V₁ × V₂ × ... × Vn φ : V₁ × V₂ × ... × Vn→ W₁ × W₂ × ... × Wn φ : V₁ × V₂ × ... × Vn→ W₁ × W₂ × ... × Wm (all being vector spaces) as in what kind of books there are discussing this, what this would be useful for etc...
 Sci Advisor P: 906 i believe that what you are looking for is "the tensor product" which is "the most general bilinear map" (with two vector spaces) or "multilinear map" (with more than two).
 P: 530 Thanks for the response, so I take it that φ : V₁ × V₂ → N is the "most general bilinear map" that can be formed because of the "universal property" [that φ is the most general bilinear map because for any other bilinear map h : V₁ × V₂ → M you have a unique linear map ω : N → M such that h = ω o φ] which if I understand it correctly says that: ω o φ : V₁ × V₂ → M | (m₁,m₂) ↦ (ω o φ)(m₁,m₂) = ω[φ(∑ᵢλᵢeᵢ,∑jμje'j)] __________________________ = ω[λ₁μ₁φ(e₁,e₁') + λ₂μ₁φ(e₂,e₁') + λ₁μ₂φ(e₁,e₂') + λ₂μ₂φ(e₂,e₂')] __________________________ = ω[λ₁μ₁φ(e₁,e₁')] + ω[λ₂μ₁φ(e₂,e₁')] + ω[λ₁μ₂φ(e₁,e₂')] + ω[λ₂μ₂φ(e₂,e₂')] __________________________ = λ₁μ₁ω[φ(e₁,e₁')] + λ₂μ₁ω[φ(e₂,e₁')] + λ₁μ₂ω[φ(e₁,e₂')] + λ₂μ₂ω[φ(e₂,e₂')] __________________________ = ∑ᵢ∑jλᵢμjω[φ(eᵢ,ej')] is linear, which itself is just a fancy way of ensuring that image elements of φ are linear because they can be mapped to image elements of a separate bilinear map h. So I take it that when you generalize this to an indexed product of vector spaces the tensor product that results is also the the most general multilinear map φ : ΠᵢVᵢ→ N?