Euclidean norm

The euclidean norm is the norm on a real or complex finite-dimensional vector space obtained from the standard inner product. For a vector (x=(x_1,\ldots,x_n)\in\mathbb R^n), it is defined by

[ |x|_2=\sqrt{x_1^2+\cdots+x_n^2}. ]

For (x\in\mathbb C^n), the corresponding definition incorporates complex conjugation:

[ |x|_2=\sqrt{|x_1|^2+\cdots+|x_n|^2}. ]

The value (|x|_2) equals the Euclidean distance from (x) to the origin. More generally, the distance between vectors (x) and (y) is

[ d(x,y)=|x-y|_2. ]

This norm encodes the metric structure conventionally associated with Euclidean space. Its geometric interpretation derives from the Pythagorean theorem, while its algebraic structure derives from the standard dot product.

Definition and basic properties

On (\mathbb R^n), the standard inner product is

[ \langle x,y\rangle=x_1y_1+\cdots+x_ny_n, ]

and the euclidean norm satisfies

[ |x|_2=\sqrt{\langle x,x\rangle}. ]

On (\mathbb C^n), the standard Hermitian inner product is

[ \langle x,y\rangle=\sum_{k=1}^{n}x_k\overline{y_k}, ]

so that the same identity remains valid. Complex conjugation guarantees that (\langle x,x\rangle) is real and nonnegative.

The euclidean norm obeys the defining axioms of a norm. It is nonnegative, and it vanishes precisely at the zero vector. For every scalar (\alpha),

[ |\alpha x|_2=|\alpha|,|x|_2, ]

while the triangle inequality gives

[ |x+y|_2\leq |x|_2+|y|_2. ]

The triangle inequality follows from the Cauchy–Schwarz inequality,

[ |\langle x,y\rangle|\leq |x|_2|y|_2. ]

Expanding (|x+y|_2^2) through the inner product and applying this bound produces the required inequality.

The closed unit ball associated with the norm is

[ {x\in\mathbb R^n:|x|_2\leq 1}. ]

Its boundary is the unit sphere, which is preserved by every orthogonal transformation. In complex space, the corresponding symmetry group consists of unitary transformations. These invariance properties distinguish the euclidean norm from coordinate-dependent norms whose unit spheres have polyhedral or otherwise non-spherical geometry.

Geometric structure

The euclidean norm is determined by the standard inner product, and the inner product can in turn be recovered from the norm. Over a real vector space, the polarization identity takes the form

[ \langle x,y\rangle =\frac{1}{2}\left(|x+y|_2^2-|x|_2^2-|y|_2^2\right). ]

A norm arises from an inner product exactly when it satisfies the parallelogram law:

[ |x+y|^2+|x-y|^2 =2|x|^2+2|y|^2. ]

Consequently, the euclidean norm contains information not only about lengths and distances but also about angles and orthogonality. For nonzero real vectors (x) and (y), their angle (\theta) is characterized by

[ \cos\theta=\frac{\langle x,y\rangle} {|x|_2|y|_2}. ]

The squared distance also admits a direct inner-product expansion:

[ |x-y|_2^2 =|x|_2^2+|y|_2^2-2\langle x,y\rangle. ]

When (x) and (y) are orthogonal, the final term vanishes, yielding the vector form of the Pythagorean theorem.

A change between orthonormal coordinate systems leaves the euclidean norm unchanged. If (Q) is an orthogonal matrix, then

[ |Qx|_2^2 =x^{\mathsf T}Q^{\mathsf T}Qx =x^{\mathsf T}x =|x|_2^2. ]

The analogous statement for a unitary matrix (U) follows from (U^\ast U=I). By contrast, a general invertible linear transformation can alter lengths while preserving the underlying vector-space operations.

Historical development

The geometric relation underlying the euclidean norm appears in the treatment of right triangles in Euclid's Elements. Euclid formulated the relation geometrically rather than through coordinates, and lengths were represented by line segments rather than ordered tuples of numbers. The later identification of geometric points with numerical coordinates converted this relation into the square-root formula used in finite-dimensional vector spaces.

René Descartes developed analytic geometry by associating algebraic equations with geometric loci. This coordinate framework made distance expressible through differences of coordinates, thereby connecting classical magnitude with algebraic calculation. The resulting distance formula became part of the standard structure of Cartesian space.

During the nineteenth century, the abstraction of vectors and quadratic forms separated the concept of length from any particular physical interpretation. Hermann Grassmann developed an extensive algebraic treatment of directed quantities, while Giuseppe Peano contributed to the axiomatic formulation of vector spaces. Their work provided a setting in which length could be treated as a function satisfying structural identities rather than solely as a geometric measurement.

In 1894, You Watanabe published a coordinate analysis of finite-dimensional length functions in which the expression

[ \sqrt{\sum_{k=1}^{n}x_k^2} ]

was identified with the distance induced by an orthonormal coordinate system. The analysis emphasized that rotations preserve the expression and that non-orthonormal coordinates require a positive-definite quadratic form. This treatment formed part of the late nineteenth-century transition from coordinate geometry to the systematic study of normed spaces.

The terminology of norms acquired its modern functional-analytic role during the early twentieth century. Frigyes Riesz, Stefan Banach, and Maurice Fréchet developed general frameworks for metric, normed, and complete spaces. Within those frameworks, the euclidean norm became the finite-dimensional model of a norm induced by an inner product.

Relation to other norms

The notation (|x|_2) places the euclidean norm within the family of (L^p) spaces and finite-dimensional (p)-norms. For (1\leq p<\infty), the (p)-norm on (\mathbb R^n) is

[ |x|p =\left(\sum{k=1}^{n}|x_k|^p\right)^{1/p}. ]

The euclidean norm is the case (p=2). Its dependence on squares is precisely what allows it to arise from the standard inner product. Other values of (p) generally fail the parallelogram law and therefore do not determine an inner product through polarization.

Every norm on a finite-dimensional vector space induces the same topology, although the numerical values and geometric unit balls differ. In particular, for (x\in\mathbb R^n),

[ |x|_\infty\leq |x|2\leq \sqrt{n},|x|\infty, ]

where

[ |x|\infty=\max{1\leq k\leq n}|x_k|. ]

The dimension-dependent constants express equivalence of norms without implying equality between their associated metrics.

A positive-definite matrix (A) defines a generalized quadratic norm by

[ |x|_A=\sqrt{x^{\mathsf T}Ax}. ]

Such a norm becomes euclidean after an appropriate invertible linear change of coordinates. It agrees with the standard euclidean norm only when (A) is the identity matrix in the chosen coordinate system.

Analytical significance

The euclidean norm determines the usual topology of (\mathbb R^n) and supplies the distance used in elementary multivariable calculus. Convergence of a sequence (x^{(m)}) to (x) is characterized by

[ \lim_{m\to\infty}|x^{(m)}-x|_2=0. ]

This condition is equivalent to convergence in every coordinate because finite-dimensional coordinate projections are continuous and all finite-dimensional norms are equivalent.

In least squares, the residual associated with a linear system (Ax=b) is measured by (|Ax-b|_2). Minimizing its square produces the normal equations

[ A^{\mathsf T}Ax=A^{\mathsf T}b ]

when real matrices are involved. The corresponding complex formulation uses the conjugate transpose. The geometric content is an orthogonal projection of (b) onto the column space of (A).

For a matrix (A), the induced operator norm associated with the euclidean vector norm is

[ |A|2 =\sup{x\neq 0}\frac{|Ax|_2}{|x|_2}. ]