Difference between revisions of "Space of all k-linear maps"
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Latest revision as of 00:08, 1 August 2015
Contents
[hide]Definition
For a k∈N and a family U1,⋯,Uk, of vector spaces over a field F we denote[1]:
- L(U1,⋯,Uk) as the space of all k-linear maps with domain U1×⋯×Uk that map to any vector space (over F)
Here (V,F) is a vector space
- L(U1,⋯,Uk;V) is the space of all k-linear maps of the form :U1×⋯×Uk→V
- Claim: L(U1,⋯,Uk;V) is a vector space. For f,g∈L(U1,⋯,Uk;V) and λ∈F we define the operations as:
- (f+g)(x1,⋯,xk)=f(x1,⋯,xk)+g(x1,⋯,xk) and
- (λf)(x1,⋯,xk)=λf(x1,⋯,xk)
- Claim: L(U1,⋯,Uk;V) is a vector space. For f,g∈L(U1,⋯,Uk;V) and λ∈F we define the operations as:
Proof of claim
[Expand]
Claim 1: L(U1,⋯,Uk;V) is a vector space with the operations (f+g)(x1,⋯,xk)=f(x1,⋯,xk)+g(x1,⋯,xk) and (λf)(x1,⋯,xk)=λf(x1,⋯,xk)
See also
References
- Jump up ↑ Multilinear Algebra - Second Edition - W. H. Greub