In mathematics, a topological algebra A over a topological field K is a topological vector space together with a continuous, bilinear multiplication
that makes it an algebra over K. A unital associative topological algebra is a topological ring. An example of a topological algebra is the algebra C[0,1] of continuous real-valued functions on the closed unit interval [0,1], or more generally any Banach algebra.
The term was coined by David van Dantzig; it appears in the title of his doctoral dissertation (1931).
The natural notion of subspace in a topological algebra is that of a (topologically) closed subalgebra. A topological algebra A is said to be generated by a subset S if A itself is the smallest closed subalgebra of A that contains S. For example by the Stone–Weierstrass theorem, the set {id[a,b]} consisting only of the identity function id[a,b] is a generating set of the Banach algebra C[a,b] when 0 < a < b.