Difference between revisions of "Algebra"

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(Created page with ": '''Note: ''' Not to be confused with an algebra of sets (as would be encountered in measure theory) see Algebra (Disambiguation) for all uses __TOC__...")
 
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: '''Note: ''' Not to be confused with [[Algebra of sets|an algebra of sets]] (as would be encountered in measure theory) see [[Algebra (Disambiguation)]] for all uses
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{{Disambiguation}}
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__TOC__
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* [[Algebra (linear algebra)]] - a vector space coupled with a notion of ''multiplication''
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==Definition==
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* [[Algebra (measure theory)]] - {{AKA}}: an [[Algebra of sets]]
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An ''algebra'' (over a [[field]] {{M|F}}) is a [[vector space]], {{M|(V,F)}}, endowed with a [[bilinear map]] used for the ''product'' operation on the vector space<ref name="ITSM">Introduction to Smooth Manifolds - John M. Lee - Second Edition - Springer GTM</ref>, that is a vector space {{M|(V,F)}} with a map:
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* [[Abstract Algebra (subject)]] - often shortened to just "algebra" - a branch of mathematics
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* {{M|1=P:V\times V\rightarrow V}}, which is [[bilinear map|bilinear]]<ref group="Recall">Recall that for a map to be bilinear we require:
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{{Definition|Measure Theory|Manifolds|Differential Geometry|Real Analysis|Linear Algebra}}
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:# {{M|1=P(\alpha x+\beta y,z)=\alpha P(x,z)+\beta P(y,z)}} and
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'''Note: ''' if you came here not expecting a disambiguation page, you probably want [[Algebra (linear algebra)]] as it was moved from this page. [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 15:39, 27 March 2016 (UTC)
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:# {{M|1=P(x,\alpha y+\beta z)=\alpha P(x,y)+\beta P(x,z)}} for all {{M|\alpha,\beta\in F}} and for all {{M|x,y\in V}}</ref> called the "product".
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* So now we define {{M|1=xy:=P(x,y)}} on the space, thus endowing our vector space with a notion of product.
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===Properties===
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We may say an ''algebra'' is any ''(zero or more)'' of the following if it satisfies the definitions:
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{| class="wikitable" border="1"
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|-
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! Property
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! Definition
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|-
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! Commutative<ref name="ITSM"/>
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| If the product is commutative. That is if {{M|1=xy=yx}} (which is the same as {{M|1=P(x,y)=P(y,x)}})
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|-
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! Associative<ref name="ITSM"/>
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| If the product is associative. That is if {{M|1=x(yz)=(xy)z}} (which is the same as {{M|1=P(x,P(y,z))=P(P(x,y),z)}})
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|}
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==Examples==
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* [[Class of smooth real-valued functions on R-n|The class of smooth real-valued functions on {{M|\mathbb{R}^n}}]]
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(See also the category: [[:Category:Examples of algebras|Examples of algebras]])
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{{Todo|{{Highlight|Investigate the product of {{M|k}}-differentiable real valued functions}}}}
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==Recall notes==
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<references group="Recall"/>
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==References==
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<references/>
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{{Definition|Manifolds|Differential Geometry|Functional Analysis}}
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Latest revision as of 15:39, 27 March 2016

Disambiguation

This page lists articles associated with the same title.

If an internal link led you here, you may wish to change the link to point directly to the intended article.


Algebra may refer to:

Note: if you came here not expecting a disambiguation page, you probably want Algebra (linear algebra) as it was moved from this page. Alec (talk) 15:39, 27 March 2016 (UTC)

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