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.
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 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 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.
The term was coined by David van Dantzig; it appears in the title of his doctoral dissertation (1931).
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.
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 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 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.
The term was coined by David van Dantzig; it appears in the title of his doctoral dissertation (1931).