In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and zero is only at the origin. In particular, the Euclidean distance in a Euclidean space is defined by a norm on the associated Euclidean vector space, called the Euclidean norm, the 2-norm, or, sometimes, the magnitude or length of the vector. This norm can be defined as the square root of the inner product of a vector with itself.

A seminorm satisfies the first two properties of a norm but may be zero for vectors other than the origin. A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a seminormed vector space.

The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm". It can also refer to a norm that can take infinite values or to certain functions parametrised by a directed set.

Given a vector space ${\displaystyle X}$ over a subfield ${\displaystyle F}$ of the complex numbers ${\displaystyle \mathbb {C} ,}$ a norm on ${\displaystyle X}$ is a real-valued function ${\displaystyle p:X\to \mathbb {R} }$ with the following properties, where ${\displaystyle |s|}$ denotes the usual absolute value of a scalar ${\displaystyle s}$:

A seminorm on ${\displaystyle X}$ is a function ${\displaystyle p:X\to \mathbb {R} }$ that has properties (1.) and (2.) so that in particular, every norm is also a seminorm (and thus also a sublinear functional). However, there exist seminorms that are not norms. Properties (1.) and (2.) imply that if ${\displaystyle p}$ is a norm (or more generally, a seminorm), then ${\displaystyle p(0)=0}$ and that ${\displaystyle p}$ also has the following property:

Some authors include non-negativity as part of the definition of "norm", although this is not necessary. Although this article defined "positive" to be a synonym of "positive definite", some authors instead define "positive" to be a synonym of "non-negative"; these definitions are not equivalent.

Suppose that ${\displaystyle p}$ and ${\displaystyle q}$ are two norms (or seminorms) on a vector space ${\displaystyle X.}$ Then ${\displaystyle p}$ and ${\displaystyle q}$ are called equivalent, if there exist two positive real constants ${\displaystyle c}$ and ${\displaystyle C}$ such that for every vector ${\displaystyle x\in X,}$ $${\displaystyle cq(x)\leq p(x)\leq Cq(x).}$$ The relation "${\displaystyle p}$ is equivalent to ${\displaystyle q}$" is reflexive, symmetric (${\displaystyle cq\leq p\leq Cq}$ implies ${\displaystyle {\tfrac {1}{C}}p\leq q\leq {\tfrac {1}{c}}p}$), and transitive and thus defines an equivalence relation on the set of all norms on ${\displaystyle X.}$ The norms ${\displaystyle p}$ and ${\displaystyle q}$ are equivalent if and only if they induce the same topology on ${\displaystyle X.}$ Any two norms on a finite-dimensional space are equivalent but this does not extend to infinite-dimensional spaces.

If a norm ${\displaystyle p:X\to \mathbb {R} }$ is given on a vector space ${\displaystyle X,}$ then the norm of a vector ${\displaystyle z\in X}$ is usually denoted by enclosing it within double vertical lines: ${\displaystyle \|z\|=p(z)}$, as proposed by Stefan Banach in his doctoral thesis from 1920. Such notation is also sometimes used if ${\displaystyle p}$ is only a seminorm. For the length of a vector in Euclidean space (which is an example of a norm, as explained below), the notation ${\displaystyle |x|}$ with single vertical lines is also widespread.