that turns into an algebra over 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: 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
- stereotype continuity: for each totally bounded set and for each neighbourhood of zero there is a neighbourhood of zero such that , or
- separate continuity: for each element and for each neighbourhood of zero there is a neighbourhood of zero such that and .
(Certanly, 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.
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