Topological Algebra

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In mathematics, a **topological algebra** *A* over a topological field **K** is a topological vector space together with a continuous 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 of continuous real-valued functions on the closed unit interval ,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 Stoneâ€“Weierstrass theorem, the set consisting only of the identity function id<sub></sub> is a generating set of the Banach algebra C.

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- <math>cdot :Atimes A longrightarrow A</math>
- <math>(a,b)longmapsto acdot b</math>

that makes it an algebra over

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

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