In mathematics, a '''topological algebra''' <math>A</math> is an algebra and at the same time a topological space, where the algebraic and the topological structures are coherent in a specified sense.
==Definition== A '''topological algebra''' <math>A</math> over a topological field <math>K</math> is a topological vector space together with a bilinear multiplication
:<math>\cdot: A \times A \to A</math>, :<math>(a,b) \mapsto a \cdot b</math>
that turns <math>A</math> into an algebra over <math>K</math> and is continuous in some definite sense. Usually the ''continuity of the multiplication'' is expressed by one of the following (non-equivalent) requirements:
* ''joint continuity'':{{sfn|Beckenstein|Narici|Suffel|1977}} for each neighbourhood of zero <math>U\subseteq A</math> there are neighbourhoods of zero <math>V\subseteq A</math> and <math>W\subseteq A</math> such that <math>V \cdot W\subseteq U</math> (in other words, this condition means that the multiplication is continuous as a map between topological spaces {{nowrap|<math>A \times A \to A</math>),}} or * ''stereotype continuity'':{{sfn|Akbarov|2003}} for each totally bounded set <math>S\subseteq A</math> and for each neighbourhood of zero <math>U\subseteq A</math> there is a neighbourhood of zero <math>V\subseteq A</math> such that <math>S \cdot V\subseteq U</math> and <math>V \cdot S\subseteq U</math>, or * ''separate continuity'':{{sfn|Mallios|1986}} for each element <math>a\in A</math> and for each neighbourhood of zero <math>U\subseteq A</math> there is a neighbourhood of zero <math>V\subseteq A</math> such that <math>a\cdot V\subseteq U</math> and <math>V\cdot a\subseteq U</math>.
(Certainly, joint continuity implies stereotype continuity, and stereotype continuity implies separate continuity.) In the first case <math>A</math> is called a "''topological algebra with jointly continuous multiplication''", and in the last, "''with separately continuous multiplication''".
A unital associative topological algebra is (sometimes) called a topological ring.
==History== The term was coined by David van Dantzig; it appears in the title of his doctoral dissertation (1931).
== Examples == :1. Fréchet algebras are examples of associative topological algebras with jointly continuous multiplication. :2. Banach algebras are special cases of Fréchet algebras. :3. Stereotype algebras are examples of associative topological algebras with stereotype continuous multiplication.
==Notes== {{reflist}}
== External links == * {{nlab|id=topological+algebra|title=Topological algebra}}
==References== * {{cite book | last1=Beckenstein | first1=E. | last2=Narici | first2=L. | last3=Suffel | first3=C. | title=Topological Algebras | publisher=North Holland | location=Amsterdam | year=1977 | isbn=9780080871356 }} *{{cite journal|last=Akbarov|first=S.S.|title=Pontryagin duality in the theory of topological vector spaces and in topological algebra|journal=Journal of Mathematical Sciences|year=2003|volume=113|issue=2|pages=179–349|doi=10.1023/A:1020929201133|s2cid=115297067|doi-access=free}} * {{cite book | last=Mallios | first=A. | title=Topological Algebras | publisher=North Holland | location=Amsterdam | year=1986 | isbn=9780080872353 }} * {{cite book | last=Balachandran | first=V.K. | title=Topological Algebras | publisher=North Holland | location=Amsterdam | year=2000 | isbn=9780080543086 }} * {{cite book | last=Fragoulopoulou | first=M. | title=Topological Algebras with Involution | publisher=North Holland | location=Amsterdam | year=2005 | isbn=9780444520258 }}
Category:Topological algebra Category:Topological vector spaces Category:Algebras
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