In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a topological algebra
is an
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
and at the same time a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
, where the algebraic and the topological structures are coherent in a specified sense.
Definition
A topological algebra
over a
topological field
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is wi ...
is a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
together with a bilinear multiplication
:
,
:
that turns
into an
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
over
and is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
in some definite sense. Usually the ''continuity of the multiplication'' is expressed by one of the following (non-equivalent) requirements:
* ''joint continuity'': for each
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...
of zero
there are neighbourhoods of zero
and
such that
(in other words, this condition means that the multiplication is continuous as a map between topological spaces or
* ''stereotype continuity'': for each
totally bounded set In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size†...
and for each neighbourhood of zero
there is a neighbourhood of zero
such that
and
, or
* ''separate continuity'': for each element
and for each neighbourhood of zero
there is a neighbourhood of zero
such that
and
.
(Certainly, joint continuity implies stereotype continuity, and stereotype continuity implies separate continuity.) In the first case
is called a "''topological algebra with jointly continuous multiplication''", and in the last, "''with separately continuous multiplication''".
A unital
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
topological algebra is (sometimes) called a
topological ring In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mode ...
.
History
The term was coined by
David van Dantzig
David van Dantzig (September 23, 1900 – July 22, 1959) was a Dutch mathematician, well known for the construction in topology of the dyadic solenoid. He was a member of the Significs Group.
Biography
Born to a Jewish family in Amsterdam in ...
; it appears in the title of his
doctoral dissertation
A thesis ( : theses), or dissertation (abbreviated diss.), is a document submitted in support of candidature for an academic degree or professional qualification presenting the author's research and findings.International Standard ISO 7144: ...
(1931).
Examples
:1.
Fréchet algebra
In mathematics, especially functional analysis, a Fréchet algebra, named after Maurice René Fréchet, is an associative algebra A over the real or complex numbers that at the same time is also a (locally convex) Fréchet space. The multiplication ...
s are examples of associative topological algebras with jointly continuous multiplication.
:2.
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach spa ...
s are special cases of
Fréchet algebra
In mathematics, especially functional analysis, a Fréchet algebra, named after Maurice René Fréchet, is an associative algebra A over the real or complex numbers that at the same time is also a (locally convex) Fréchet space. The multiplication ...
s.
:3.
Stereotype algebra In mathematics, a topological algebra A 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 A over a topological field K is a ...
s are examples of associative topological algebras with stereotype continuous multiplication.
Notes
External links
*
References
*
*
*
*
*
Topological vector spaces
Algebras
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