Topological group

In mathematics, topological groups are groups and topological spaces at the same time, where the group operations are required to be continuous. This connects these two structures together, relating them to each other.

Topological groups were studied extensively in the period of 1925 to 1940. Haar and Weil (respectively in 1933 and 1940) showed that the integrals and Fourier series are special cases of a construction that can be defined on a very wide class of topological groups.

Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example, in physics. In functional analysis, every topological vector space is an additive topological group with the additional property that scalar multiplication is continuous; consequently, many results from the theory of topological groups can be applied to functional analysis.

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