Algebra
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- Note: Not to be confused with an algebra of sets (as would be encountered in measure theory) see Algebra (Disambiguation) for all uses
Definition
An algebra (over a field F) is a vector space, (V,F), endowed with a bilinear map used for the product operation on the vector space[1], that is a vector space (V,F) with a map:
- P:V×V→V, which is bilinear[Recall 1] called the "product".
- So now we define xy:=P(x,y) on the space, thus endowing our vector space with a notion of product.
Properties
We may say an algebra is any (zero or more) of the following if it satisfies the definitions:
| Property | Definition |
|---|---|
| Commutative[1] | If the product is commutative. That is if xy=yx (which is the same as P(x,y)=P(y,x)) |
| Associative[1] | If the product is associative. That is if x(yz)=(xy)z (which is the same as P(x,P(y,z))=P(P(x,y),z)) |
Examples
(See also the category: Examples of algebras)
TODO: Investigate the product of k-differentiable real valued functions
Recall notes
- Jump up ↑ Recall that for a map to be bilinear we require:
- P(αx+βy,z)=αP(x,z)+βP(y,z) and
- P(x,αy+βz)=αP(x,y)+βP(x,z) for all α,β∈F and for all x,y∈V
References
- ↑ Jump up to: 1.0 1.1 1.2 Introduction to Smooth Manifolds - John M. Lee - Second Edition - Springer GTM
