The completion functor is a mathematical tool that takes a topological algebra and 'completes' it with respect to a specific topology, often the topology induced by a seminorm or a family of seminorms. This process results in a new algebra that is complete, meaning every Cauchy sequence converges within the algebra. Completion functors play an essential role in the study of functional analysis and noncommutative geometry, providing a way to construct larger algebras that retain the properties of their original structures while allowing for convergence and continuity.
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