Topological algebra


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.

Definition

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 requirements:
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 called a topological ring.

History

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

Examples