Ring of sets: Difference between revisions

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{{Noref|date=November 2009}}
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In [[mathematics]], a '''topological algebra''' ''A'' over a [[topological field]] '''K''' is a [[topological vector space]] together with a continuous multiplication
 
:<math>\cdot :A\times A \longrightarrow A</math>
:<math>(a,b)\longmapsto a\cdot b</math>
 
that makes it an [[algebra over a field|algebra]] over '''K'''. A unital [[associative algebra|associative]] topological algebra is a [[topological ring]].
An example of a topological algebra is the algebra C[0,1] of continuous real-valued functions on the closed unit interval [0,1],
or more generally any [[Banach algebra]].
 
The term was coined by [[David van Dantzig]]; it appears in the title of his [[Thesis|doctoral dissertation]] (1931).
 
The natural notion of subspace in a topological algebra is that of a (topologically) closed [[subalgebra]]. A topological algebra ''A'' is said to be generated by a subset ''S'' if ''A'' itself is the smallest closed subalgebra of ''A'' that contains ''S''. For example by the [[Stone–Weierstrass theorem]], the set {id<sub>[0,1]</sub>} consisting only of the identity function id<sub>[0,1]</sub> is a generating set of the Banach algebra C[0,1].
 
[[Category:Topological vector spaces]]
[[Category:Topological algebra]]
[[Category:Algebras]]
 
{{topology-stub}}

Revision as of 14:35, 15 February 2014

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