Concept

Normed vector space

Summary
In mathematics, a normed vector space or normed space is a vector space 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 V is a vector space over K, where K is a field equal to \mathbb R or to \mathbb C, then a norm on V is a map V\to\mathbb R, typically denoted by \lVert\cdot \rVert, satisfying the following four axioms: #Non-negativity: for every x\in V,; \lVert x \rVert \ge 0. #Positive definiteness: for every x \in V, ; \lVert x\rVert=0 if and only if x is the zero vector.

Absolute homogeneity: for every \lambda\in K and x\in V,\lVert \lambda x \rVert = |\lambda|, \lVert x\rVert

Triangle inequality: for every x\in V and y\in V

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