# Norm (mathematics)

In mathematics, a norm is a real-valued function on a vector space that assigns each vector a length or magnitude, subject to three axioms: only the zero vector has length zero, scaling a vector scales its length by the absolute value of the scalar, and the length of a sum is at most the sum of the lengths. A vector space equipped with a norm is called a normed vector space, or normed space.<sup>[1](https://sites.engineering.ucsb.edu/~bamieh/pubs_new/MM_251219.pdf)</sup><sup> • </sup><sup>[2](https://encyclopediaofmath.org/wiki/Norm)</sup>

| Key fact | Detail |
|---|---|
| Defining axioms | Positivity (\|\|u\|\| ≥ 0, zero only at the zero vector), homogeneity (\|\|αu\|\| = \|α\| \|\|u\|\|), and the triangle inequality (\|\|u + v\|\| ≤ \|\|u\|\| + \|\|v\|\|)<sup>[3](https://maths-people.anu.edu.au/john/Assets/Lecture%20Notes/B21H_97.pdf)</sup> |
| Induced metric | Every norm defines a distance d(x, y) = \|\|x − y\|\|, making the space a metric space with translation-invariant, homogeneous distances<sup>[2](https://encyclopediaofmath.org/wiki/Norm)</sup><sup> • </sup><sup>[3](https://maths-people.anu.edu.au/john/Assets/Lecture%20Notes/B21H_97.pdf)</sup> |
| p-norms on R<sup>n</sup> | For 1 ≤ p < ∞, \|\|(x₁, …, xₙ)\|\|ₚ = (Σ|xᵢ|ᵖ)^(1/p); the sup norm is max\{|x₁|, …, |xₙ|\}, and \|\|x\|\|ₚ → \|\|x\|\|∞ as p → ∞<sup>[3](https://maths-people.anu.edu.au/john/Assets/Lecture%20Notes/B21H_97.pdf)</sup> |
| Finite-dimensional fact | On any finite-dimensional real or complex vector space, all norms are equivalent<sup>[4](https://www.cfm.brown.edu/people/dobrush/cs52/Mathematica/Part4/norm.html)</sup> |
| Completeness | A normed space complete in its induced metric is a Banach space; every normed space has a Banach completion<sup>[2](https://encyclopediaofmath.org/wiki/Norm)</sup> |
| Dual space | The continuous linear functionals on a normed space X form a Banach space under \|\|ψ\|\| = sup\{|ψ(x)| : \|\|x\|\| ≤ 1\}<sup>[2](https://encyclopediaofmath.org/wiki/Norm)</sup> |
| Boundary case | The expression (Σ|xᵢ|ᵖ)^(1/p) fails the triangle inequality for 0 < p < 1 and is a quasi-norm, not a norm<sup>[5](https://math.stackexchange.com/questions/4575695/existence-of-non-trivial-bounded-linear-functionals-with-a-quasi-norm)</sup> |

## Definition

A norm on a vector space V over the real or complex numbers is a function \|\|·\|\| : V → ℝ satisfying three axioms for all vectors and all scalars α:<sup>[1](https://sites.engineering.ucsb.edu/~bamieh/pubs_new/MM_251219.pdf)</sup>

1. **Definiteness (positivity).** \|\|v\|\| = 0 if and only if v is the zero vector, and \|\|v\|\| ≥ 0 otherwise.<sup>[3](https://maths-people.anu.edu.au/john/Assets/Lecture%20Notes/B21H_97.pdf)</sup>
2. **Homogeneity.** \|\|αv\|\| = \|α\| \|\|v\|\|, so scaling a vector by α scales its length by the magnitude of α.<sup>[1](https://sites.engineering.ucsb.edu/~bamieh/pubs_new/MM_251219.pdf)</sup>
3. **Triangle inequality.** \|\|v + w\|\| ≤ \|\|v\|\| + \|\|w\|\|; equivalently \|\|v − w\|\| ≤ \|\|v\|\| + \|\|w\|\|, since substituting −w for w and using homogeneity gives the two forms interchangeably.<sup>[1](https://sites.engineering.ucsb.edu/~bamieh/pubs_new/MM_251219.pdf)</sup>

The triangle inequality yields two frequently used consequences: \|\|u\|\| ≤ \|\|u − v\|\| + \|\|v\|\|, and the reverse triangle inequality \|\|\|u\|\| − \|\|v\|\|\|\| ≤ \|\|u − v\|\|, which bounds how much the lengths of two vectors can differ by the distance between them.<sup>[3](https://maths-people.anu.edu.au/john/Assets/Lecture%20Notes/B21H_97.pdf)</sup>

A vector space may carry many different norms, but not every vector space is normable.<sup>[6](https://wiki.math.ntnu.no/linearmethods/basicspaces/normedvectorspaces?s%5B%5D=no)</sup> The absolute value on ℝ, or complex magnitude on ℂ, is the model case of a norm, and when ℝ or ℂ is treated as a normed space this is the norm meant unless stated otherwise.<sup>[7](https://commeunjeu.org/cours/fr-mpi-normed-vector-spaces/?lang=en)</sup><sup> • </sup><sup>[8](https://controldynsys.github.io/131-wi25site/assets/Lectures_on_Linear_Systems_Theory.pdf)</sup>

## The induced metric and topology

Every norm induces a metric by the formula dist(x, y) = \|\|x − y\|\|, and hence a topology compatible with that metric, giving the space the structure of a topological vector space.<sup>[2](https://encyclopediaofmath.org/wiki/Norm)</sup> The proof mirrors the Euclidean case: positivity and symmetry are immediate, and the triangle inequality for distances follows from the norm's triangle inequality applied to x − y = (x − z) + (z − y).<sup>[3](https://maths-people.anu.edu.au/john/Assets/Lecture%20Notes/B21H_97.pdf)</sup>

Norm-induced metrics have two properties general metrics lack: they are translation invariant, since d(x + a, y + a) = d(x, y), and homogeneous, since d(αx, αy) = \|α\| d(x, y).<sup>[1](https://sites.engineering.ucsb.edu/~bamieh/pubs_new/MM_251219.pdf)</sup>

## Examples

**Euclidean and p-norms.** On ℝⁿ the [Euclidean norm](https://www.edgepedia.org/euclidean-norm) is

\[ \|(x_1, \ldots, x_n)\|_2 = \left( (x_1)^2 + \cdots + (x_n)^2 \right)^{1/2}. \]

Its triangle inequality is the only non-obvious axiom to verify, and it holds because ℝⁿ with the dot product is an inner product space and the Euclidean norm is the norm corresponding to that inner product; the triangle inequality holds for the norm of any inner product space.<sup>[3](https://maths-people.anu.edu.au/john/Assets/Lecture%20Notes/B21H_97.pdf)</sup> More generally, for 1 ≤ p < ∞,

\[ \|(x_1, \ldots, x_n)\|_p = \left( \sum_{i=1}^{n} |x_i|^p \right)^{1/p} \]

defines the p-norm, and \|\|(x_1, \ldots, x_n)\|\|_\infty = max\{|x_1|, \ldots, |x_n|\} defines the sup norm, with \|\|x\|\|ₚ → \|\|x\|\|∞ as p → ∞.<sup>[3](https://maths-people.anu.edu.au/john/Assets/Lecture%20Notes/B21H_97.pdf)</sup> [The 1](https://www.edgepedia.org/the-1)-norm corresponds to Manhattan or taxicab distance, in which the direct path between two points is not necessarily the shortest route.<sup>[9](https://www.youtube.com/watch?v=vECP7Dw7TL8)</sup>

**Function spaces.** The space C[a, b] of continuous real-valued functions on an interval [a, b] carries several norms: the sup norm \|\|f\|\|∞ = sup\{|f(x)| : a ≤ x ≤ b\}, which for continuous functions is attained and could be written as a maximum; the 1-norm \|\|f\|\|₁ = ∫ₐᵇ |f|; and the 2-norm (\∫ₐᵇ f²)^(1/2), whose triangle inequality again follows from the inner product structure.<sup>[3](https://maths-people.anu.edu.au/john/Assets/Lecture%20Notes/B21H_97.pdf)</sup> The bounded sequences ℓ∞(ℕ) with the sup norm, and its subspace c₀ of sequences converging to 0, are further examples.<sup>[3](https://maths-people.anu.edu.au/john/Assets/Lecture%20Notes/B21H_97.pdf)</sup>

**What is not a norm.** For 0 < p < 1 the expression (Σ|xᵢ|ᵖ)^(1/p) violates subadditivity and is therefore not a norm; such functionals are called quasi-norms.<sup>[5](https://math.stackexchange.com/questions/4575695/existence-of-non-trivial-bounded-linear-functionals-with-a-quasi-norm)</sup>

## Completeness and Banach spaces

A normed space that is complete in its induced metric, meaning every [Cauchy sequence](https://www.edgepedia.org/cauchy-sequence) converges, is called a [Banach space](https://www.edgepedia.org/banach-space), and every normed space has a Banach completion.<sup>[2](https://encyclopediaofmath.org/wiki/Norm)</sup> The name honors [Stefan Banach](https://www.edgepedia.org/stefan-banach), who in 1922 began a systematic study of these spaces based on axioms he introduced himself.<sup>[10](https://encyclopediaofmath.org/wiki/Banach_space)</sup> The function spaces introduced by David Hilbert, Maurice Fréchet, and Frigyes Riesz between 1904 and 1918 served as the starting point for the theory.<sup>[10](https://encyclopediaofmath.org/wiki/Banach_space)</sup>

The dual space of a normed space, the collection X* of continuous linear functionals ψ on X, is itself a Banach space under the norm

\[ \|\psi\| = \sup\{ |\psi(x)| : \|x\| \leq 1 \}. \]

The norms of all functionals are attained at points of the unit ball of the original space if and only if the space is reflexive.<sup>[2](https://encyclopediaofmath.org/wiki/Norm)</sup>

## Finite-dimensional theory

On a finite-dimensional real or complex vector space, any two norms are equivalent.<sup>[4](https://www.cfm.brown.edu/people/dobrush/cs52/Mathematica/Part4/norm.html)</sup> Equivalence means each norm bounds the other up to constant factors, so all norms on such a space induce the same topology and the same notions of convergence and continuity.<sup>[11](https://pages.cs.wisc.edu/~ruixuan/math522/defs.html)</sup>

Finite dimensionality also underlies operator theory: for a linear operator T : V → W between normed spaces, continuity holds if and only if there exists a constant C ≥ 0 such that \|\|Tv\|\|_W ≤ C\|\|v\|\|_V for every v.<sup>[11](https://pages.cs.wisc.edu/~ruixuan/math522/defs.html)</sup><sup> • </sup><sup>[19](https://faculty.curgus.wwu.edu/Courses/528_201620/528.html)</sup>

## Norms in computation and statistics

**Induced matrix norms.** When a linear map A acts between normed vector spaces V and W, the induced norm is

\[ \|A\|_{\mathcal{V},\mathcal{W}} = \sup_{v \neq 0} \frac{\|Av\|_{\mathcal{W}}}{\|v\|_{\mathcal{V}}} = \sup_{\|v\|_{\mathcal{V}} = 1} \|Av\|_{\mathcal{W}}. \]

When V is finite-dimensional the unit ball is compact and the supremum is attained.<sup>[12](https://www.cs.cornell.edu/courses/cs4220/2026sp/lec/2026-01-30.html)</sup> Induced norms satisfy the submultiplicative properties \|\|Av\|\| ≤ \|\|A\|\| \|\|v\|\| and \|\|AB\|\| ≤ \|\|A\|\| \|\|B\|\|.<sup>[12](https://www.cs.cornell.edu/courses/cs4220/2026sp/lec/2026-01-30.html)</sup> The matrix 1-norm and ∞-norm, induced by the vector 1-norm and ∞-norm, are easy to compute as maximum absolute column sums and maximum absolute row sums respectively, \|\|A\|\|₁ = maxⱼ Σᵢ |aᵢⱼ| and \|\|A\|\|∞ = maxᵢ Σⱼ |aᵢⱼ|; the matrix 2-norm is useful but not easy to compute.<sup>[12](https://www.cs.cornell.edu/courses/cs4220/2026sp/lec/2026-01-30.html)</sup> These norms support practical estimates such as the [Neumann series](https://www.edgepedia.org/neumann-series) bound \|\|(I − F)⁻¹\|\| ≤ 1/(1 − \|\|F\|\|) when \|\|F\|\| < 1, which quantifies how a small perturbation of a matrix changes the norm of its inverse.<sup>[12](https://www.cs.cornell.edu/courses/cs4220/2026sp/lec/2026-01-30.html)</sup>

**Regularization.** Norms serve as penalty terms in statistical estimation. The lasso, proposed by [Robert Tibshirani](https://www.edgepedia.org/robert-tibshirani), a statistician at the [University of Toronto](https://www.edgepedia.org/university-of-toronto), minimizes the residual sum of squares subject to the sum of the absolute values of the coefficients (an ℓ¹ constraint) being bounded by a constant; this constraint tends to produce some coefficients that are exactly 0, giving interpretable models.<sup>[13](https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.2517-6161.1996.tb02080.x)</sup> Mixed ℓ¹/ℓq norms with q > 1 encourage group sparsity in regression and classification models.<sup>[14](https://arxiv.org/abs/1009.4766)</sup> In low-rank matrix learning, the nuclear norm is the widely used convex regularizer, though experimental surveys report that non-convex regularizers can provide a large advantage over it.<sup>[15](https://arxiv.org/abs/1808.04521)</sup>

## Related and generalized structures

Several weakenings of the norm concept appear in the literature. A semi-norm drops the requirement that \|\|x\|\| = 0 only at the zero vector; quotient constructions turn semi-normed spaces into normed ones.<sup>[16](https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/pages/lecture-notes-and-readings/)</sup> Asymmetric norms, which need not satisfy \|\|−x\|\| = \|\|x\|\|, lead to dual spaces that are convex cones rather than linear spaces, requiring reworking of duality results such as the open mapping and closed graph theorems.<sup>[17](https://arxiv.org/abs/1006.1175)</sup> In constructive analysis, q-normed spaces, introduced by D. L. Johns under the name pseudonormed spaces, use families of seminorms to handle spaces such as L∞ in which not every element is constructively normable.<sup>[18](https://arxiv.org/abs/2606.25236)</sup>

## References

1. [Bassam Bamieh, Notes on Linear Algebra and Functional Analysis, Section 2.2, Normed Vector Spaces, UC Santa Barbara](https://sites.engineering.ucsb.edu/~bamieh/pubs_new/MM_251219.pdf)
2. [Norm, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Norm)
3. [Lecture notes, Normed Vector Spaces, Mathematical Sciences Institute, Australian National University](https://maths-people.anu.edu.au/john/Assets/Lecture%20Notes/B21H_97.pdf)
4. [Linear Algebra, Part 1: Norm and distance, Brown University](https://www.cfm.brown.edu/people/dobrush/cs52/Mathematica/Part4/norm.html)
5. [Existence of non-trivial bounded linear functionals with a quasi-norm, Mathematics Stack Exchange](https://math.stackexchange.com/questions/4575695/existence-of-non-trivial-bounded-linear-functionals-with-a-quasi-norm)
6. [Normed spaces, NTNU Linear Methods wiki](https://wiki.math.ntnu.no/linearmethods/basicspaces/normedvectorspaces?s%5B%5D=no)
7. [Normed vector spaces, CommeUnJeu course notes](https://commeunjeu.org/cours/fr-mpi-normed-vector-spaces/?lang=en)
8. [Lectures on Linear Systems Theory, Section 1.3](https://controldynsys.github.io/131-wi25site/assets/Lectures_on_Linear_Systems_Theory.pdf)
9. [Norms, Norm Vector Spaces: definition and examples, YouTube lecture by F. Maalouf](https://www.youtube.com/watch?v=vECP7Dw7TL8)
10. [Banach space, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Banach_space)
11. [MATH 522 (Analysis II): Definitions and Theorems, University of Wisconsin](https://pages.cs.wisc.edu/~ruixuan/math522/defs.html)
12. [David Bindel, Norms revisited, CS 4220: Numerical Analysis, Cornell University](https://www.cs.cornell.edu/courses/cs4220/2026sp/lec/2026-01-30.html)
13. [Robert Tibshirani (1996), Regression Shrinkage and Selection Via the Lasso, Journal of the Royal Statistical Society: Series B, 58: 267–288](https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.2517-6161.1996.tb02080.x)
14. [Efficient L1/Lq Norm Regularization, arXiv](https://arxiv.org/abs/1009.4766)
15. [Low Rank Regularization: A Review, arXiv](https://arxiv.org/abs/1808.04521)
16. [Lecture Notes and Readings, 18.102 Introduction to Functional Analysis, MIT OpenCourseWare](https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/pages/lecture-notes-and-readings/)
17. [Functional analysis in asymmetric normed spaces, arXiv](https://arxiv.org/abs/1006.1175)
18. [Douglas S. Bridges, Notes on q-normed Spaces in Constructive Analysis, arXiv](https://arxiv.org/abs/2606.25236)
19. [Math 528 - Spring 2016](https://faculty.curgus.wwu.edu/Courses/528_201620/528.html)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Linear and multilinear algebra › Vector spaces and linear maps*

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