Normed space
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Normed space or normed vector space is the vector space in which a norm is defined. A normed vector space is a pair (V, ‖·‖ ) where V is a vector space and ‖·‖ a norm on V. A seminormed vector space is a pair (V,p) where V is a vector space and p a seminorm on V. In order to be normed we have to define a norm from a distance, so a normed space is a metric space.
A example of normed vector spaces are the Hilbert space and the Bannach space.