Difference between revisions of "Algebra"

Revision as of 23:40, 21 October 2015

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 [ilmath]F[/ilmath]) is a vector space, [ilmath](V,F)[/ilmath], endowed with a bilinear map used for the product operation on the vector space^[1], that is a vector space [ilmath](V,F)[/ilmath] with a map:

• [ilmath]P:V\times V\rightarrow V[/ilmath], which is bilinear^{[Recall 1]} called the "product".
• So now we define [ilmath]xy:=P(x,y)[/ilmath] 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:

Examples

• The class of smooth real-valued functions on [ilmath]\mathbb{R}^n[/ilmath]

(See also the category: Examples of algebras)

TODO: Investigate the product of [ilmath]k[/ilmath]-differentiable real valued functions

Recall notes

1. ↑ Recall that for a map to be bilinear we require:

   1. [ilmath]P(\alpha x+\beta y,z)=\alpha P(x,z)+\beta P(y,z)[/ilmath] and
   2. [ilmath]P(x,\alpha y+\beta z)=\alpha P(x,y)+\beta P(x,z)[/ilmath] for all [ilmath]\alpha,\beta\in F[/ilmath] and for all [ilmath]x,y\in V[/ilmath]

References

1. ↑ ^{1.0 1.1 1.2} Introduction to Smooth Manifolds - John M. Lee - Second Edition - Springer GTM