that turns into an algebra over and is continuous in a definite sense. Usually (but not always) the continuity of the multiplication is expressed by one of the following two (non-equivalent) requirements:
- joint continuity: 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: 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.
- 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.
- Beckenstein, E.; Narici, L.; Suﬀel, C. (1977). Topological Algebras. Amsterdam: North Holland. ISBN 9780080871356.
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