Norm - Knowlpedia

A norm on a vector space $V$ over $\mathbb F=\mathbb R$ or $\mathbb C$ is a function $\|\cdot\|:V\to[0,\infty)$ such that for all $u,v\in V$ and $a\in\mathbb{F}$ :

\|v\|=0\iff v=0,\qquad \|a v\|=|a|\,\|v\|,\qquad \|u+v\|\le \|u\|+\|v\|.

Here $|a|$ denotes the absolute value for real scalars and modulus for complex scalars.

Remarks

A norm induces a metric by $d(u,v)=\|u-v\|$ , making $V$ into a metric space and thus giving notions of convergence and continuity.

Examples

The norm records length, while the induced metric records separation: $d(u,v)=\|u-v\|$ . Different norms can give different lengths on the same vector space, even when they induce the same topology in finite dimensions.

On $\mathbb{R}^n$ , the Euclidean norm $\|x\|_2=\sqrt{\sum_{i=1}^n x_i^2}$ is a norm.
On $\mathbb{R}^n$ , the $\ell^1$ norm $\|x\|_1=\sum_{i=1}^n |x_i|$ and the max norm $\|x\|_\infty=\max_i |x_i|$ are norms.
On continuous functions on $[0,1]$ , the sup norm $\|f\|_\infty=\sup_{t\in[0,1]} |f(t)|$ is a norm.