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Normed vector space - Wikipedia

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In mathematics, a normed vector space or normed space is a vector space, typically over the real or complex numbers, on which a norm is defined.[1] A norm is a generalization of the intuitive notion of "length" in the physical world. If 𝑉 is a vector space over 𝐾 , where 𝐾 is a field equal to 𝑅 or to 𝐶 , then a norm on 𝑉 is a map 𝑉 → 𝑅 , typically denoted by ‖ ⋅ ‖ , satisfying the following four axioms: If 𝑉 is a real or complex vector space as above, and ‖ ⋅ ‖ is a norm on 𝑉 , then the ordered pair ( 𝑉 , ‖ ⋅ ‖ ) is called a normed vector space. If it is clear from context which norm is intended, then it is common to denote the normed vector space simply by 𝑉 . A norm induces a distance, called its (norm) induced metric, by the formula 𝑑 ( 𝑥 , 𝑦 ) = ‖ 𝑦 − 𝑥 ‖ . which makes any normed vector space into a metric space and a topological vector space. If this metric space is complete then the normed space is a Banach space. Every normed vector space can

Normed vector space - Wikipedia Jump to content From Wikipedia, the free encyclopedia Vector space on which a distance is defined This article includes a list of general references but lacks corresponding inline citations . Please help improve this article by introducing more precise citations. ( December 2019 ) ( Learn how and when to remove this message ) Hierarchy of mathematical spaces. Inner product spaces are a subset of normed vector spaces, which are a subset of metric spaces , which in turn are a subset of topological spaces . In mathematics , a normed vector space or normed space is

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