Norm (mathematics) · aiwiki.page

A norm is a nonnegative real-valued function that measures the size of an element of a vector space. It generalizes ordinary length to objects such as coordinate vectors, matrices, and functions. Unlike an arbitrary measure of size, a norm must respect scalar multiplication and addition and vanish only at the zero vector. Norms connect linear algebra with mathematical analysis by providing a way to define distance and convergence. (ocw.mit.edu)

Definition and basic properties

Let $V$ be a vector space over the real numbers or complex numbers. A norm is a map

\|\cdot\|:V\longrightarrow[0,\infty)

satisfying three conditions for all $x,y\in V$ and scalars $\alpha$ :

Definiteness: $\|x\|=0$ if and only if $x=0$ .
Absolute homogeneity: $\|\alpha x\|=|\alpha|\|x\|$ .
Triangle inequality: $\|x+y\|\leq\|x\|+\|y\|$ .

The pair $(V,\|\cdot\|)$ is a normed vector space. The axioms imply symmetry, $\|-x\|=\|x\|$ , and the reverse triangle inequality:

\bigl|\|x\|-\|y\|\bigr|\leq\|x-y\|.

Consequently, the norm function is continuous in its induced distance. (tbetcke.github.io)

A seminorm retains nonnegativity, absolute homogeneity, and the triangle inequality but may vanish on nonzero vectors. For example, $p(x_1,x_2)=|x_1|$ ignores the second coordinate and is therefore not a norm on $\mathbb R^2$ . (ocw.mit.edu)

Coordinate norms and geometry

For $x=(x_1,\ldots,x_n)\in\mathbb R^n$ or $\mathbb C^n$ , the $p$ -norms are

\|x\|_p=\left(\sum_{i=1}^{n}|x_i|^p\right)^{1/p}, \qquad 1\leq p<\infty,

with

\|x\|_\infty=\max_i|x_i|.

Important cases are the $1$ -norm, which sums absolute coordinate values; the $2$ -norm, or Euclidean norm; and the infinity norm, which measures the largest coordinate magnitude. The Euclidean norm gives ordinary Euclidean distance through $\|x-y\|_2$ . (tbetcke.github.io)

Different norms give different geometries. In $\mathbb R^2$ , their unit balls are respectively a diamond, a disk, and an axis-aligned square for $p=1,2,\infty$ . Every norm is a convex function, and its unit ball is a convex set. These properties follow from homogeneity and the triangle inequality. (stanford.edu)

For $0<p<1$ , the same coordinate formula does not define a norm in dimensions at least two because the triangle inequality fails. The expression commonly called the “ $0$ -norm,” counting nonzero coordinates, is also not a norm: multiplying a nonzero vector by a nonzero scalar leaves that count unchanged, violating homogeneity. (stanford.edu)

Distance, convergence, and equivalence

Every norm defines a metric space by

d(x,y)=\|x-y\|.

This distance is translation invariant. Convergence means $\|x_k-x\|\to0$ . A normed space is a Banach space when every Cauchy sequence converges to an element of the space. Completeness is an additional property, not part of the norm axioms. (web.math.princeton.edu)

Two norms are equivalent if positive constants $c,C$ exist such that

c\|x\|_a\leq\|x\|_b\leq C\|x\|_a

for every $x$ . Equivalent norms define the same topology and the same convergent sequences. All norms on a fixed finite-dimensional real or complex vector space are equivalent. This does not mean that their numerical values or unit balls coincide. In infinite-dimensional spaces, norms need not be equivalent. (math.mit.edu)

Inner-product norms

An inner product induces a norm by

\|x\|=\sqrt{\langle x,x\rangle}.

The Cauchy–Schwarz inequality establishes the triangle inequality for this construction. Not every norm arises from an inner product. A real or complex norm does so exactly when it satisfies the parallelogram law:

\|x+y\|^2+\|x-y\|^2 =2\|x\|^2+2\|y\|^2.

When this condition holds, polarization recovers the unique compatible inner product. A complete inner-product space is a Hilbert space. Thus Hilbert spaces are Banach spaces with additional geometric structure. (ocw.mit.edu)

Function spaces

Norms also measure functions. On a measure space $(X,\mu)$ , the $L^p$ norm is

\|f\|_p=\left(\int_X|f|^p\,d\mu\right)^{1/p}, \qquad 1\leq p<\infty.

The space contains functions for which this quantity is finite. Its elements are equivalence classes identifying functions equal almost everywhere; this identification ensures definiteness. For $p=\infty$ , the norm is the essential supremum, which disregards exceptional sets of measure zero. (math.ucdavis.edu)

For a continuous function on a compact interval,

\|f\|_\infty=\max_x|f(x)|.

Convergence in this supremum norm is uniform convergence. By contrast, integral norms measure aggregate discrepancies and can permit substantial differences on sets of small measure. These distinctions are central in functional analysis. (math.ucdavis.edu)

Matrix norms, dual norms, and applications

A matrix $A$ , regarded as a linear map, has an induced operator norm

\|A\|=\sup_{x\ne0}\frac{\|Ax\|}{\|x\|}.

For Euclidean vector norms, this is the largest singular value. The Frobenius norm instead measures the matrix as an array:

\|A\|_F=\left(\sum_{i,j}|a_{ij}|^2\right)^{1/2}.

It generally differs from the induced Euclidean norm. Compatible induced norms satisfy $\|AB\|\leq\|A\|\|B\|$ . (tbetcke.github.io)

On $\mathbb R^n$ , the dual norm is

\|y\|_*=\sup_{\|x\|\leq1}|y^{\mathsf T}x|.

The dual of the $p$ -norm is the $q$ -norm when $1/p+1/q=1$ , including the pairing of $1$ and infinity. In mathematical optimization, norms measure residual errors and constrain feasible solutions. Norm penalties also provide regularization: $1$ -norm penalties can promote sparse solutions, while squared $2$ -norm penalties control overall coefficient magnitude. (stanford.edu)