PlanetMath: normed vector space

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normed vector space

(Definition)

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

$\lVert v \rVert \geq 0$ for all $v\in V$ and $\lVert v \rVert =0$ if and only if in (positive definiteness)
$\lVert \lambda v \rVert = \lvert \lambda \rvert \lVert v \rVert$ for all $v\in V$ and all $\lambda\in\mathbb{F}$
$\lVert v+w \rVert \leq\lVert v \rVert +\lVert w \rVert$ for all $v,w\in V$ (the triangle inequality)

The function $\lVert \cdot \rVert$ is called a norm on .

Some properties of norms:

If is a subspace of then can be made into a normed space by simply restricting the norm on to . This is called the induced norm on .
Any normed vector space $(V,\lVert \cdot \rVert )$ is a metric space under the metric $d\colon V \times V \to \mathbb{R}$ given by $d(u,v)=\lVert u-v \rVert$ . This is called the metric induced by the norm $\lVert \cdot \rVert$ .
It follows that any normed space is a locally convex topological vector space, in the topology induced by the metric defined above.
In this metric, the norm defines a continuous map from to $\mathbb{R}$ - this is an easy consequence of the triangle inequality.
If $(V, \langle , \rangle )$ is an inner product space, then there is a natural induced norm given by $\lVert v \rVert = \sqrt{\langle v , v \rangle }$ for all $v \in V$ .
The norm is a convex function of its argument.