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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. A norm is a generalization of the intuitive notion of "length" in the physical world. If is a vector space over , where is a field equal to or to , then a norm on is a map , typically denoted by , satisfying the following four axioms:Non-negativity: for every ,.
Positive definiteness: for every , if and only if is the zero vector.
Absolute homogeneity: for every and ,
Triangle inequality: for every and ,
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