Ring of sets: Difference between revisions
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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 18:44, 26 February 2013
Template:Noref In mathematics, a topological algebra A over a topological field K is a topological vector space together with a continuous multiplication
that makes it an algebra over K. A unital 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 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[0,1]} consisting only of the identity function id[0,1] is a generating set of the Banach algebra C[0,1].