A normed algebra is an algebra over a field with a norm satisfying . Its completion is a Banach algebra, since multiplication extends continuously. Completeness is additional structure, rather than part of the definition of a normed algebra.
A normed division algebra is a nonzero unital normed algebra in which every nonzero element has a two-sided multiplicative inverse. The complex Gelfand-Mazur theorem forces such an algebra to be , even when completeness is not initially assumed: embed it in its Banach algebra completion and use nonemptiness of the Banach-algebra spectrum.
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A **normed algebra** is a specific type of algebraic structure that combines features of both normed spaces and algebras. To qualify as a normed algebra, a mathematical object must meet the following criteria: 1. **Algebra over a field**: A normed algebra \( A \) is a vector space over a field \( F \) (typically the field of real or complex numbers) equipped with a multiplication operation that is associative and distributive with respect to vector addition.