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

1. $\norm{v}\geq 0$ for all $v\in V$ and $\norm{v}=0$ if and only if $v=0$ in $V$ (positive definiteness)
2. $\norm{\lambda v} = \abs{\lambda}\norm{v}$ for all $v\in V$ and all $\lambda\in\mathbb{F}$
3. $\norm{v+w}\leq\norm{v}+\norm{w}$ for all $v,w\in V$ (the triangle inequality)

The function $\norm{\cdot}$ 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 $(V,\norm{\cdot})$ is a metric space under the metric $d\colon V \times V \to \mathbb{R}$ given by $d(u,v)=\norm{u-v}$ . This is called the metric induced by the norm $\norm{\cdot}$ .
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 $(V, \ip{}{})$ is an inner product space, then there is a natural induced norm given by $\norm{v} = \sqrt{\ip{v}{v}}$ for all $v \in V$ .
6. The norm is a convex function of its argument.