normed vector space
Let $\mathbb{F}$ be a field which is either $\mathbb{R}$ or $\u2102$. A over $\mathbb{F}$ is a pair $(V,\parallel \cdot \parallel )$ where $V$ is a vector space over $\mathbb{F}$ and $\parallel \cdot \parallel :V\to \mathbb{R}$ is a function such that

1.
$\parallel v\parallel \ge 0$ for all $v\in V$ and $\parallel v\parallel =0$ if and only if $v=0$ in $V$ (positive definiteness)

2.
$\parallel \lambda v\parallel =\lambda \parallel v\parallel $ for all $v\in V$ and all $\lambda \in \mathbb{F}$

3.
$\parallel v+w\parallel \le \parallel v\parallel +\parallel w\parallel $ for all $v,w\in V$ (the triangle inequality^{})
The function $\parallel \cdot \parallel $ is called a norm on $V$.
Some properties of norms:

1.
If $W$ is a subspace^{} of $V$ then $W$ can be made into a normed space^{} by simply restricting the norm on $V$ to $W$. This is called the induced norm on $W$.

2.
Any normed vector space $(V,\parallel \cdot \parallel )$ is a metric space under the metric $d:V\times V\to \mathbb{R}$ given by $d(u,v)=\parallel uv\parallel $. This is called the metric induced by the norm $\mathrm{\parallel}\mathrm{\cdot}\mathrm{\parallel}$.

3.
It follows that any normed space is a locally convex topological vector space, in the topology^{} induced by the metric defined above.

4.
In this metric, the norm defines a continuous map^{} from $V$ to $\mathbb{R}$  this is an easy consequence of the triangle inequality.

5.
If $(V,\u27e8,\u27e9)$ is an inner product space^{}, then there is a natural induced norm given by $\parallel v\parallel =\sqrt{\u27e8v,v\u27e9}$ for all $v\in V$.

6.
The norm is a convex function of its argument.
Title  normed vector space 
Canonical name  NormedVectorSpace 
Date of creation  20130322 12:13:45 
Last modified on  20130322 12:13:45 
Owner  rspuzio (6075) 
Last modified by  rspuzio (6075) 
Numerical id  14 
Author  rspuzio (6075) 
Entry type  Definition 
Classification  msc 46B99 
Synonym  normed space 
Synonym  normed linear space 
Related topic  CauchySchwarzInequality 
Related topic  VectorNorm 
Related topic  PseudometricSpace 
Related topic  MetricSpace 
Related topic  UnitVector 
Related topic  ProofOfGramSchmidtOrthogonalizationProcedure 
Related topic  EveryNormedSpaceWithSchauderBasisIsSeparable 
Related topic  EveryNormedSpaceWithSchauderBasisIsSeparable2 
Related topic  FrobeniusProduct 
Defines  norm 
Defines  metric induced by a norm 
Defines  metric induced by the norm 
Defines  induced norm 