Topological algebra
In mathematics, a topological algebra 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 over a topological field is a topological vector space together with a bilinear multiplication
- ,
that turns into an algebra over and is continuous in a definite sense. Usually (but not always[1]) the continuity of the multiplication is expressed by one of the following two (non-equivalent) requirements:
- joint continuity[2]: for each neighbourhood 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
- separate continuity[3]: for each element and for each neighbourhood of zero there is a neighbourhood of zero such that and .
In the first case is called a topological algebra with jointly continuous multiplication, and in the second - 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 separately continuous multiplication.
External links
Notes
References
- Beckenstein, E.; Narici, L.; Suffel, C. (1977). Topological Algebras. Amsterdam: North Holland. ISBN 9780080871356.
- Mallios, A. (1986). Topological Algebras. Amsterdam: North Holland. ISBN 9780080872353.