Normed vector space

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a normed vector space or normed space is a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> over the <a href="/facts/Real_number/R02gw5Pb">real</a> or <a href="/facts/Complex_number/2w7mqI7e">complex</a> numbers on which a <a href="/facts/Norm_(mathematics)/xIbR4uE1">norm</a> is defined. A norm is a generalization of the intuitive notion of "length" in the physical world. If 
 
 
 
 V
 
 
 {\displaystyle V}
 
 is a vector space over 
 
 
 
 K
 
 
 {\displaystyle K}
 
, where 
 
 
 
 K
 
 
 {\displaystyle K}
 
 is a field equal to 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
 or to 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
, then a norm on 
 
 
 
 V
 
 
 {\displaystyle V}
 
 is a map 
 
 
 
 V
 →
 
 R
 
 
 
 {\displaystyle V\to \mathbb {R} }
 
, typically denoted by 
 
 
 
 ‖
 ⋅
 ‖
 
 
 {\displaystyle \lVert \cdot \rVert }
 
, satisfying the following four axioms:

<ol><li>Non-negativity: for every 
 
 
 
 x
 ∈
 V
 
 
 {\displaystyle x\in V}
 
,
 
 
 
 
 ‖
 x
 ‖
 ≥
 0
 
 
 {\displaystyle \;\lVert x\rVert \geq 0}
 
.</li>
<li>Positive definiteness: for every 
 
 
 
 x
 ∈
 V
 
 
 {\displaystyle x\in V}
 
, 
 
 
 
 
 ‖
 x
 ‖
 =
 0
 
 
 {\displaystyle \;\lVert x\rVert =0}
 
 if and only if 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is the zero vector.</li>
<li>Absolute homogeneity: for every 
 
 
 
 λ
 ∈
 K
 
 
 {\displaystyle \lambda \in K}
 
 and 
 
 
 
 x
 ∈
 V
 
 
 {\displaystyle x\in V}
 
,
 
 
 
 ‖
 λ
 x
 ‖
 =
 
 |
 
 λ
 
 |
 
 
 ‖
 x
 ‖
 
 
 {\displaystyle \lVert \lambda x\rVert =|\lambda |\,\lVert x\rVert }
 
</li>
<li><a href="/facts/Triangle_inequality/KubLjkwr">Triangle inequality</a>: for every 
 
 
 
 x
 ∈
 V
 
 
 {\displaystyle x\in V}
 
 and 
 
 
 
 y
 ∈
 V
 
 
 {\displaystyle y\in V}
 
,
 
 
 
 ‖
 x
 +
 y
 ‖
 ≤
 ‖
 x
 ‖
 +
 ‖
 y
 ‖
 .
 
 
 {\displaystyle \|x+y\|\leq \|x\|+\|y\|.}
 
</li></ol>
If 
 
 
 
 V
 
 
 {\displaystyle V}
 
 is a real or complex vector space as above, and 
 
 
 
 ‖
 ⋅
 ‖
 
 
 {\displaystyle \lVert \cdot \rVert }
 
 is a norm on 
 
 
 
 V
 
 
 {\displaystyle V}
 
, then the ordered pair 
 
 
 
 (
 V
 ,
 ‖
 ⋅
 ‖
 )
 
 
 {\displaystyle (V,\lVert \cdot \rVert )}
 
 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
 
 
 {\displaystyle V}
 
.
A norm induces a <a href="/facts/Metric_(mathematics)/RkUptSo8">distance</a>, called its <a href="/facts/Norm_induced_metric/RkUptSo8">(norm) induced metric</a>, by the formula

d
        (
        x
        ,
        y
        )
        =
        ‖
        y
        −
        x
        ‖
        .
      
    
    {\displaystyle d(x,y)=\|y-x\|.}

which makes any normed vector space into a <a href="/facts/Metric_space/gnBBRXtc">metric space</a> and a <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a>. If this metric space is <a href="/facts/Complete_metric_space/lsckmU8G">complete</a> then the normed space is a <a href="/facts/Banach_space/sDEIk7EC">Banach space</a>. 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 <a href="/facts/Finite_sequence/gV2S4uqp">finite sequences</a> of real numbers can be normed with the <a href="/facts/Euclidean_norm/R2UbzmzM">Euclidean norm</a>, but it is not complete for this norm. 
An <a href="/facts/Inner_product_space/HoyjElEy">inner product space</a> is a normed vector space whose norm is the square root of the inner product of a vector and itself. The <a href="/facts/Euclidean_norm/R2UbzmzM">Euclidean norm</a> of a <a href="/facts/Euclidean_vector_space/R2UbzmzM">Euclidean vector space</a> is a special case that allows defining <a href="/facts/Euclidean_distance/9qDoQKQe">Euclidean distance</a> by the formula

d
        (
        A
        ,
        B
        )
        =
        ‖
        
          
            
              A
              B
            
            →
          
        
        ‖
        .
      
    
    {\displaystyle d(A,B)=\|{\overrightarrow {AB}}\|.}

The study of normed spaces and Banach spaces is a fundamental part of <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, a major subfield of mathematics.

Normed vector space open-in-new

Normed vector space