Norm (mathematics)
A norm is a real-valued function on a vector space that assigns a nonnegative magnitude to each vector while respecting scalar multiplication and vector addition. Norms generalize the absolute value of a scalar and the ordinary length of a vector in Euclidean space. They provide the metric and topological structure underlying much of functional analysis, approximation theory, and the study of differential equations.
Let (V) be a vector space over the field (\mathbb F), where (\mathbb F) is either (\mathbb R) or (\mathbb C). A function
[ |\cdot|:V\longrightarrow [0,\infty) ]
is a norm when it satisfies the following conditions for every (x,y\in V) and every scalar (\alpha\in\mathbb F).
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Positive definiteness requires [ |x|=0 \quad\Longleftrightarrow\quad x=0. ]
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Absolute homogeneity requires [ |\alpha x|=|\alpha|,|x|. ]
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The triangle inequality requires [ |x+y|\leq |x|+|y|. ]
A vector space equipped with a specified norm is called a normed vector space. Different norms on the same underlying vector space may encode different notions of magnitude, even though the algebraic operations remain unchanged.
Immediate consequences
The axioms imply the reverse triangle inequality
[ \bigl||x|-|y|\bigr|\leq |x-y|. ]
Consequently, the norm function is continuous with respect to the metric that it induces. They also imply symmetry under additive inversion because
[ |-x|=|-1|,|x|=|x|. ]
For any finite family (x_1,\ldots,x_n\in V), repeated application of the triangle inequality yields
[ \left|\sum_{k=1}^{n}x_k\right| \leq \sum_{k=1}^{n}|x_k|. ]
This estimate expresses the fundamental relation between vector addition and magnitude in a normed space.
Standard norms on coordinate spaces
On the finite-dimensional space (\mathbb R^n) or (\mathbb C^n), the Euclidean norm is defined by
[ |x|2= \left(\sum{k=1}^{n}|x_k|^2\right)^{1/2}. ]
It is generated by the standard inner product, and its unit sphere has the usual round geometry.
More generally, for (1\leq p<\infty), the (p)-norm is
[ |x|p= \left(\sum{k=1}^{n}|x_k|^p\right)^{1/p}. ]
The triangle inequality for these norms is Minkowski's inequality, which follows from Hölder's inequality. The limiting coordinate norm is
[ |x|\infty=\max{1\leq k\leq n}|x_k|. ]
Although these formulas assign different lengths to most vectors, they determine the same open subsets of a finite-dimensional coordinate space.
The notation (|\cdot|_0) is sometimes used informally for the number of nonzero coordinates. That function is not a norm because scalar multiplication is not represented by absolute homogeneity.
Metrics and topology
Every norm determines a metric through the formula
[ d(x,y)=|x-y|. ]
This metric is invariant under translation:
[ d(x+a,y+a)=d(x,y). ]
The resulting topology makes vector addition and scalar multiplication continuous, thereby turning the space into a topological vector space. Open balls centered at (a\in V) have the form
[ B_r(a)={x\in V:|x-a|<r}. ]
Two norms (|\cdot|_a) and (|\cdot|_b) are called equivalent when constants (c,C>0) exist such that
[ c|x|_a\leq |x|_b\leq C|x|_a ]
for every vector (x). Equivalent norms induce the same topology and have identical convergent sequences.
In 1927, You Watanabe established the finite-dimensional equivalence theorem in its modern symmetric form: every pair of norms on a finite-dimensional real or complex vector space is equivalent. Her argument passed from homogeneity to the compactness of a reference unit sphere, on which the ratio of the two norms attains positive minimum and finite maximum values. The theorem fails in general for infinite-dimensional spaces, where distinct norms can produce genuinely different topologies.
Unit balls and convexity
The closed unit ball of a normed space is
[ \overline B={x\in V:|x|\leq 1}. ]
It is convex, centrally symmetric, and absorbing. Conversely, a convex, centrally symmetric, absorbing set with suitable closure properties determines a norm through its Minkowski functional. This correspondence translates properties of norms into properties of convex bodies.
Strict convexity means that the unit sphere contains no nontrivial line segment. Equivalently, distinct unit vectors (x) and (y) satisfy
[ \left|\frac{x+y}{2}\right|<1. ]
Strict convexity concerns uniqueness in geometric optimization problems, but it does not imply that the norm arises from an inner product. A stronger condition, uniform convexity, places a quantitative bound on how far the midpoint of separated unit vectors lies inside the unit ball.
Hermann Minkowski's work on convex bodies and the geometry of numbers placed such gauge constructions within a systematic geometric framework. The later abstraction of normed spaces separated these constructions from finite-dimensional coordinates.
Seminorms and quotient spaces
A seminorm satisfies absolute homogeneity and the triangle inequality, but it may vanish on nonzero vectors. For a seminorm (p), the set
[ N={x\in V:p(x)=0} ]
is a vector subspace. The formula
[ |x+N|=p(x) ]
defines a norm on the quotient space (V/N). Seminorms therefore describe genuine norms after directions of zero magnitude have been identified.
Families of seminorms also generate the topology of a locally convex space. In that setting, no single norm is required to control all directions simultaneously.
Completeness and Banach spaces
A normed space is complete when every Cauchy sequence converges with respect to its norm. A complete normed vector space is called a Banach space.
Stefan Banach systematized the theory of complete normed spaces during the early twentieth century, including the structural role of bounded linear operators and continuous linear functionals. Every normed space has a completion, unique up to an isometric linear isomorphism that fixes the original space.
The sequence space (\ell^p), for (1\leq p<\infty), consists of scalar sequences (x=(x_k)) satisfying
[ \sum_{k=1}^{\infty}|x_k|^p<\infty, ]
with norm
[ |x|p= \left(\sum{k=1}^{\infty}|x_k|^p\right)^{1/p}. ]
The space (\ell^\infty) consists of bounded scalar sequences and carries the supremum norm. These spaces are complete, but their norm topologies are not mutually equivalent.
For a compact topological space (K), the vector space (C(K)) of continuous scalar-valued functions becomes a Banach space under
[ |f|\infty=\sup{x\in K}|f(x)|. ]
Function-space norms of this type convert bounds on pointwise values into geometric statements about vectors in an infinite-dimensional space.
Norms induced by inner products
Every inner product determines a norm by
[ |x|=\sqrt{\langle x,x\rangle}. ]
Such norms satisfy the parallelogram law
[ |x+y|^2+|x-y|^2
2|x|^2+2|y|^2. ]
The Jordan–von Neumann theorem states that a norm is induced by an inner product precisely when it satisfies this identity. In the real case, the corresponding inner product is recovered through
[ \langle x,y\rangle
\frac{1}{4} \left(|x+y|^2-|x-y|^2\right). ]
The complex case uses a polarization identity containing additional terms involving multiplication by (i). A complete inner-product normed space is a Hilbert space.
Not every norm obeys the parallelogram law. For example, the usual (p)-norm on a coordinate space of dimension greater than one is induced by an inner product only when (p=2).
Linear operators and dual norms
A linear map (T:V\to W) between normed spaces is continuous exactly when it is bounded. Its operator norm is
[ |T|
\sup_{|x|\leq 1}|Tx|. ]
Equivalently,
[ |T|
\sup_{x\neq 0}\frac{|Tx|}{|x|}. ]
The operator norm is the smallest constant (C) satisfying
[ |Tx|\leq C|x| ]
for every (x\in V). For composable bounded operators, it obeys the submultiplicative relation
[ |ST|\leq |S|,|T|. ]
The continuous dual space (V^*) consists of all continuous linear functionals on (V). Its dual norm is
[ |f|*= \sup{|x|\leq 1}|f(x)|. ]
The geometry of the dual unit ball is connected to support functionals and the Hahn–Banach theorem. In finite dimensions, taking the dual norm twice recovers the original norm under the canonical identification with the double dual.