In mathematics, a topological algebra A 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 A over a topological field K is a topological vector space together with a bilinear multiplication that turns A into an algebra over K 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 A 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).

Examples