In words...

A norm is a function used to measure the size of a mathematical object, such as a vector, a matrix or a function.

Various types of norms can be defined depending on the particular properties of the object we want to measure. The most common norm is the Euclidean norm of vectors used for instance to compute the Euclidean distance between two points.

Each norm gives rise to a metric (a distance measure) computed as the norm of the difference between two points.

In pictures...

In maths...

A norm $\|\cdot\|$ is a function from a vector space $\X$ to the set of positive real numbers $[0,+\infty)$ which satisfies

  1. $\|x\| = 0$ if and only if $x = 0$
  2. $\|\lambda x\| = |\lambda| \|x\|$ for all scalars $\lambda$ and all vectors $x\in\X$
  3. $\|x + x^\prime\| \leq \|x\| + \|x^\prime\|$ for all $(x,x^\prime) \in\X^2$

Any norm induces a metric as $$ d(x, x^\prime) = \|x - x^\prime\| . $$

Norms in $\R^d$

The most common norms in $\R^d$ are $\ell_p$-norms defined for $p\in[1,+\infty)$ as $$ \|\g x\|_p = \left(\sum_{i=1}^{d} |x_i|^p \right)^{\frac{1}{p}} $$ with $x_i$ the $i$th component of $\g x\in\R^d$. For $p=2$, we get the Euclidean norm, which is the one induced by the standard dot product in $\R^d$: $$ \|\g x\|_2 = \sqrt{\inner{\g x}{\g x}} $$

The infinite-norm is defined as $$ \|\g x\|_{\infty} = \max_{i\in\{1,\dots,d\}} |x_i| $$

Functional norms

Functional norms can be defined. For real-valued functions $f : \X \rightarrow \R$, the $L^p$-norm is defined as $$ \|f\|_{L^p} = \left(\int_{\X} \left|f(x)\right|^p \ dx \right)^{\frac{1}{p}} $$ The essential supremum...