In a normed vector space, one can define absolute convergence as convergence of the series of norms ( \ Sigma | b _ k | ).
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The term is most commonly used for the initial topology of a topological vector space ( such as a normed vector space ) with respect to its continuous dual.
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Then is a normed vector space ( and in particular, it is a metric space . ) All metric spaces have Hausdorff completions, so let be its completion.
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:The Q might be why we should be concerned with normed vector spaces when there are nicer ones ( Euclidean spaces with infinite-dimensional versions if needed ).
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Using the bilinear map, semi norms can be constructed to define a polar topology on the vector spaces and turn them into locally convex spaces, generalizations of normed vector spaces.
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This version of the triangle inequality reduces to the one stated above in case of normed vector spaces where a metric is induced via, with being the vector pointing from point to.
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Isometries are often used in constructions where one space is closed subset of some normed vector space and that every complete metric space is isometrically isomorphic to a closed subset of some Banach space.
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When the underlying vector space X is a ( not necessarily finite-dimensional ) normed vector space, analytic questions, irrelevant in the finite-dimensional case, need to be considered.
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A proof of this theorem for separable normed vector spaces was published in 1932 by Stefan Banach, and the first proof for the general case was published in 1940 by the mathematician Leonidas Alaoglu.
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*PM : every finite dimensional normed vector space is a Banach space, id = 6633-- WP guess : every finite dimensional normed vector space is a Banach space-- Status:
