Topological algebra

In mathematics, a topological algebra A {displaystyle 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. In mathematics, a topological algebra A {displaystyle 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. A topological algebra A {displaystyle A} over a topological field K {displaystyle K} is a topological vector space together with a bilinear multiplication that turns A {displaystyle A} into an algebra over K {displaystyle K} and is continuous in a definite sense. Usually (but not always) the continuity of the multiplication is expressed by one of the following two (non-equivalent) requirements: In the first case A {displaystyle A} is called a topological algebra with jointly continuous multiplication, and in the second - 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).