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vector norm (Definition)

A vector norm on the real vector space $ V$ is a function $ f : V \to \mathbb{R}$ that satisfies the following properties:


$\displaystyle f(x) = 0 \iff x = 0$      
$\displaystyle f(x) \ge 0$   $\displaystyle x \in V$  
$\displaystyle f(x+y) \leq f(x)+f(y)$   $\displaystyle x,y \in V$  
$\displaystyle f(\alpha x) = \vert\alpha\vert f(x)$   $\displaystyle \alpha \in \mathbb{R},x\in V$  

Such a function is denoted as $ \vert\vert\,x\,\vert\vert$. Particular norms are distinguished by subscripts, such as $ \vert\vert\,x\,\vert\vert _V$, when referring to a norm in the space $ V$. A unit vector with respect to the norm $ \vert\vert\,\cdot\,\vert\vert$ is a vector $ x$ satisfying $ \vert\vert\,x\,\vert\vert = 1$.

A vector norm on a complex vector space is defined similarly.

A common (and useful) example of a real norm is the Euclidean norm given by $ \vert\vert x\vert\vert=(x_1^2 + x_2^2 + \cdots + x_n^2)^{1/2}$ defined on $ V=\mathbb{R}^n$. Note, however, that there exists vector spaces which are metric, but upon which it is not possible to define a norm. If it possible, the space is called a normed vector space. Given a metric on the vector space, a necessary and sufficient condition for this space to be a normed space, is

$\displaystyle d(x+a,y+a)=$ $\displaystyle d(x,y)$ $\displaystyle \forall x,y,a \in V$  
$\displaystyle d(\alpha x,\alpha y)=$ $\displaystyle \vert\alpha\vert d(x,y)$ $\displaystyle \forall x,y \in V, \alpha \in \mathbb{R}$  

But given a norm, a metric can always be defined by the equation $ d(x,y)=\vert\vert x-y\vert\vert$. Hence every normed space is a metric space.



"vector norm" is owned by mike. [ full author list (3) | owner history (2) ]
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See Also: Euclidean vector, metric, norm, vector p-norm, normed vector space, matrix norm, matrix p-norm, Frobenius matrix norm, Cauchy-Schwarz inequality, metric space, vector space, $L^p$-space, operator norm, bounded operator, seminorm, Banach space, Hilbert space, unit vector

Also defines:  normed vector space
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Cross-references: metric space, equation, necessary and sufficient, metric, Euclidean norm, complex, vector, unit vector, subscripts, norms, properties, function, vector space, real
There are 24 references to this entry.

This is version 16 of vector norm, born on 2001-10-06, modified 2004-03-02.
Object id is 91, canonical name is VectorNorm.
Accessed 19307 times total.

Classification:
AMS MSC46B20 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Geometry and structure of normed linear spaces)

Pending Errata and Addenda
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metric norm? by mps on 2004-02-10 16:29:51
Is the entry ``metric norm'' necessary? It seems like its content could easily be incorporated in this entry.
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