Norm (mathematics)

Lore & Background

A norm on a vector space X over a subfield F of the complex numbers is a real-valued function p: X → R with three properties: subadditivity (p(x+y) ≤ p(x)+p(y) for all x,y in X), absolute homogeneity (p(sx)=|s|p(x) for all scalars s), and positive definiteness (if p(x)=0 then x=0). A seminorm satisfies the first two properties but may be zero for vectors other than the origin. The term pseudonorm has been used as a synonym of seminorm, or to refer to a norm that can take infinite values, or to certain functions parametrised by a directed set. Note that not every vector space admits a norm—a norm requires an absolute value on the scalar field, which fields like finite fields lack.

Reader's Guide

The concept of a norm is fundamental in functional analysis and geometry, providing a rigorous way to measure the size of vectors. The Euclidean norm, defined as the square root of the inner product of a vector with itself, is a key example. Norms allow the definition of normed vector spaces, which generalize Euclidean spaces to infinite dimensions. Every vector space admits a norm, for instance via a Hamel basis. The absolute value is a norm on the real or complex numbers, and on the complex numbers it corresponds to the Euclidean norm in two dimensions. The distinction between norms and seminorms is important: seminorms lack the point-separating property, and a vector space with a seminorm is called a seminormed vector space. The properties of norms ensure they behave like distances from the origin, making them essential for analysis and geometry.

Did You Know?

• A norm is zero only at the origin, while a seminorm may be zero for other vectors.
• The Euclidean norm is defined as the square root of the inner product of a vector with itself.
• Not every vector space admits a norm; a norm requires an absolute value on the scalar field, which not all fields (e.g., finite fields) possess.

The Three Pillars of a Norm

A norm, in the language of linear algebra, is a real-valued function assigned to a vector space over a subfield of the complex numbers. Its entire identity rests on three axioms that together make it behave like a generalized distance from the origin. The first axiom, subadditivity, demands that the norm of a sum never exceeds the sum of the norms—this is the triangle inequality in functional form. The second, absolute homogeneity, requires that scaling a vector by any scalar s stretches its norm by exactly the absolute value of s, so the function commutes cleanly with scalar multiplication. The third, positive definiteness, insists that the only vector whose norm vanishes is the zero vector itself. From the first two axioms, non-negativity follows automatically: every vector maps to a value at least zero. Some authors choose to state non-negativity as an explicit fourth condition, while others treat it as a derived consequence. The interplay of these three properties is what separates a true norm from weaker structures and gives normed vector spaces their geometric rigidity.

The Euclidean Norm as Archetype

Among all possible norms, the one most students encounter first is the Euclidean norm, which underpins the familiar notion of distance in ordinary Euclidean space. It is also called the 2-norm, the magnitude of a vector, or simply its length. What makes this particular norm special is its deep connection to the inner product: the Euclidean norm of a vector can be computed as the square root of the inner product of that vector with itself. In this way, the entire geometric intuition of how long an arrow is gets encoded in a purely algebraic operation. The Euclidean norm satisfies all three defining axioms—subadditivity, absolute homogeneity, and positive definiteness—making it a canonical example of the general concept. In notation, the length of a vector in Euclidean space is frequently written with single vertical bars, distinguishing it visually from the double-bar notation used for more general norms. This special status means that whenever someone speaks of the norm without further qualification in an introductory context, they are almost certainly referring to this inner-product-derived measure of vector size.

Seminorms and the Pseudonorm Question

Relaxing just one of the three norm axioms produces a seminorm: a function that still obeys subadditivity and absolute homogeneity but is permitted to assign zero to vectors other than the origin. Because every norm automatically satisfies the seminorm conditions, the class of norms sits neatly inside the class of seminorms, though the inclusion is strict—there exist seminorms that fail positive definiteness and therefore are not norms. A vector space equipped with a chosen seminorm is termed a seminormed vector space, paralleling the normed vector space in the full case. The terminology around related concepts grows murkier with the word pseudonorm, which has been used in at least three distinct senses in the literature. In some texts it is simply a synonym for seminorm. In others it designates a norm that is allowed to take infinite values. Still other authors apply it to certain functions parametrised by a directed set. This polysemy makes the pseudonorm label one of the more ambiguous terms in functional analysis, and readers must always check the author's convention before drawing conclusions.

Banach's Notation and Terminological Quirks

The standard way to write the norm of a vector—enclosing it in double vertical bars, as in the expression for the norm of z—was proposed by Stefan Banach in his 1920 doctoral thesis. Before that convention took hold, no single typographic symbol dominated the literature. Today, the double-bar notation is so entrenched that it appears even when the underlying function is merely a seminorm rather than a full norm. For the specific case of Euclidean length, single vertical bars remain widespread, creating a visual distinction between the general and the particular. Beyond notation, the literature harbours small terminological disagreements. Some authors replace the positive-definiteness axiom with an equivalent if-and-only-if formulation, noting that the norm of zero already follows from absolute homogeneity. Others debate whether the word positive should mean positive definite, i.e. zero only at the origin, or simply non-negative, meaning every value is at least zero. These two readings are not equivalent, and the choice affects how the axioms are stated. Such minor divergences remind readers that even a deceptively simple definition can carry interpretive weight.