Editing Normed vector space (section)

{{short description|Vector space on which a distance is defined}}
{{more footnotes|date=December 2019}}
[[File:Mathematical Spaces.png|thumb|250px|Hierarchy of mathematical spaces. [[Inner product space]]s are a subset of normed vector spaces,
which are a subset of [[metric space]]s, which in turn are a subset of [[topological space]]s.]]

In [[mathematics]], a '''normed vector space''' or '''normed space''' is a [[vector space]] over the [[Real number|real]] or [[Complex number|complex]] numbers on which a [[Norm (mathematics)|norm]] is defined.<ref name="text">{{cite book|first=Frank M.|last=Callier|title=Linear System Theory|location=New York |publisher=Springer-Verlag|year=1991|isbn=0-387-97573-X}}</ref> A norm is a generalization of the intuitive notion of "length" in the physical world. If <math>V</math> is a vector space over <math>K</math>, where <math>K</math> is a field equal to <math>\mathbb R</math> or to <math>\mathbb C</math>, then a norm on <math>V</math> is a map <math>V\to\mathbb R</math>, typically denoted by <math>\lVert\cdot \rVert</math>, satisfying the following four axioms:

#Non-negativity: for every <math>x\in V</math>,<math>\; \lVert x \rVert \ge 0</math>.
#Positive definiteness: for every <math>x \in V</math>, <math>\; \lVert x\rVert=0</math> if and only if <math>x</math> is the zero vector.
# Absolute homogeneity: for every <math>\lambda\in K</math> and <math>x\in V</math>,<math display="block">\lVert \lambda x \rVert = |\lambda|\, \lVert x\rVert </math>
# [[Triangle inequality]]: for every <math>x\in V</math> and <math>y\in V</math>,<math display="block">\|x+y\| \leq \|x\| + \|y\|.</math>

If <math>V</math> is a real or complex vector space as above, and <math>\lVert\cdot\rVert</math> is a norm on <math>V</math>, then the ordered pair <math>(V,\lVert\cdot \rVert)</math> 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 <math>V</math>.

A norm induces a [[Metric (mathematics)|distance]], called its {{em|[[Norm induced metric|(norm) induced metric]]}}, by the formula
<math display="block">d(x,y) = \|y-x\|.</math>
which makes any normed vector space into a [[metric space]] and a [[topological vector space]]. If this metric space is [[Complete metric space|complete]] then the normed space is a <em>[[Banach space]]</em>. Every normed vector space can be "uniquely extended" to a Banach space, which makes normed spaces intimately related to Banach spaces. Every Banach space is a normed space but converse is not true. For example, the set of the [[finite sequence]]s of real numbers can be normed with the [[Euclidean norm]], but it is not complete for this norm. 

An [[inner product space]] is a normed vector space whose norm is the square root of the inner product of a vector and itself. The [[Euclidean norm]] of a [[Euclidean vector space]] is a special case that allows defining [[Euclidean distance]] by the formula
<math display=block>d(A, B) = \|\overrightarrow{AB}\|.</math>

The study of normed spaces and Banach spaces is a fundamental part of [[functional analysis]], a major subfield of mathematics.