Topological algebra

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


 * $$\cdot: A \times A \to A$$,
 * $$(a,b) \mapsto a \cdot b$$

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:


 * joint continuity: for each neighbourhood of zero $$U\subseteq A$$ there are neighbourhoods of zero $$V\subseteq A$$ and $$W\subseteq A$$ such that $$V \cdot W\subseteq U$$ (in other words, this condition means that the multiplication is continuous as a map between topological spaces $A \times A \to A$), or
 * stereotype continuity: for each totally bounded set $$S\subseteq A$$ and for each neighbourhood of zero $$U\subseteq A$$ there is a neighbourhood of zero $$V\subseteq A$$ such that $$S \cdot V\subseteq U$$ and $$V \cdot S\subseteq U$$, or
 * separate continuity: for each element $$a\in A$$ and for each neighbourhood of zero $$U\subseteq A$$ there is a neighbourhood of zero $$V\subseteq A$$ such that $$a\cdot V\subseteq U$$ and $$V\cdot a\subseteq U$$.

(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

 * 1. Fréchet algebras are examples of associative topological algebras with jointly continuous multiplication.
 * 2. Banach algebras are special cases of Fréchet algebras.
 * 3. Stereotype algebras are examples of associative topological algebras with stereotype continuous multiplication.