mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a topological algebra

A

is an

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

and at the same time a

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

, where the algebraic and the topological structures are coherent in a specified sense.

Definition

A topological algebra

A

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. A field is thus a fundamental algebraic structure which is widel ...

K

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 :

\cdot: A \times A \to A

, :

(a,b) \mapsto a \cdot b

that turns

A

into an

over

K

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 (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neighbourh ...

of zero

U\subseteq A

there are neighbourhoods of zero

V\subseteq A

and

W\subseteq A

such that

V \cdot W\subseteq U

(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

S\subseteq A

and for each neighbourhood of zero

U\subseteq A

there is a neighbourhood of zero

V\subseteq A

such that

S \cdot V\subseteq U

and

V \cdot S\subseteq U

, or * ''separate continuity'': for each element

a\in A

and for each neighbourhood of zero

U\subseteq A

there is a neighbourhood of zero

V\subseteq A

such that

a\cdot V\subseteq U

and

V\cdot a\subseteq U

. (Certainly, joint continuity implies stereotype continuity, and stereotype continuity implies separate continuity.) In the first case

A

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 that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...

topological algebra is (sometimes) called a

topological ring In mathematics, a topological ring is a ring R that is also a topological space such that both the addition and the multiplication are continuous as maps: R \times R \to R where R \times R carries the product topology. That means R is an additive ...

History

The term was coined by David van Dantzig; 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: D ...

(1931).

Examples

:1. Fréchet algebras 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 sp ...

s are special cases of Fréchet algebras. :3. Stereotype algebras are examples of associative topological algebras with stereotype continuous multiplication.

Notes

External links

References

* * * * * Topological vector spaces Algebras {{topology-stub