Generalizations
Several important spaces in functional analysis, for instance the space of all infinitely often differentiable functions R → R or the space of all distributions on R, are complete but are not normed vector spaces and hence not Banach spaces. In Fréchet spaces one still has a complete metric, while LF-spaces are complete uniform vector spaces arising as limits of Fréchet spaces.
Read more about this topic: Banach Space
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