Topological algebra

In mathematics, a topological algebra formula_1 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 formula_1 over a topological field formula_3 is a topological vector space together with a bilinear multiplication
that turns formula_1 into an algebra over formula_3 and is continuous in some definite sense. Usually the "continuity of the multiplication" is expressed by one of the following (non-equivalent) requirements:
(Certainly, joint continuity implies stereotype continuity, and stereotype continuity implies separate continuity.) In the first case formula_1 is called a ""topological algebra with jointly continuous multiplication", and in the last, "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).