2025-01-12

947: Multilinear Map

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definition of multilinear map

Topics


About: module

The table of contents of this article

Starting Context

Target Context


  • The reader will have a definition of multilinear map.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

Main Body

1: Structured Description 1


Here is the rules of Structured Description.

Entities:
J: ∈{ the possibly uncountable index sets }
R: ∈{ the rings }
{Mj|j∈J}: ⊆{ the R modules }
×j∈JMj: = the product module 
M: ∈{ the R modules }
∗f: :×j∈JMj→M
//

Conditions:
∀m∈×j∈JMj such that m(l)=r′ml′+r″ml″ for any l∈J, any r′,r″∈R, and any ml′,ml″∈Ml(f(m)=r′f(m′)+r″f(m″) where m′∈×j∈JMj is m′(l)=ml′ and m′(j)=m(j) for each j≠l and m″∈×j∈JMj is m″(l)=ml″ and m″(j)=m(j) for each j≠l)
//

2: Structured Description 2


Here is the rules of Structured Description.

Entities:
R: ∈{ the rings }
{M1,...,Mk}: ⊆{ the R modules }
M1×...×Mk: = the product module 
M: ∈{ the R modules }
∗f: :M1×...×Mk→M
//

Conditions:
∀m∈M1×...×Mk such that m=(m1,...,r′ml′+r″ml″,...,mk) for any l∈J, any r′,r″∈R, and any ml′,ml″∈Ml(f(m)=r′f(m′)+r″f(m″) where m′=(m1,...,ml′,...,mk) and m″=(m1,...,ml″,...,mk))
//

References
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