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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 Stone–Weierstrass theorem, the set consisting only of the identity function id() is a generating set of the Banach algebra C(). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Topological algebra」の詳細全文を読む スポンサード リンク
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