Difference between revisions of "Algebra"

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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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:# {{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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Revision as of 15:38, 27 March 2016

Disambiguation

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Algebra may refer to:

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