Normed vector space
A normed vector space, also called a normed linear space, is a vector space equipped with a function that assigns a nonnegative length to every vector and is compatible with the linear structure. The norm determines a metric, and therefore a topology, through the distance between two vectors. This combination permits algebraic operations, geometric concepts, and analytic limits to be treated within a common framework.
Normed vector spaces include finite-dimensional coordinate spaces with their usual notions of length, as well as infinite-dimensional spaces whose vectors are functions, sequences, or operators. Their systematic study forms a central part of functional analysis. When the metric induced by the norm is complete, the space is called a Banach space.
Definition
Let (V) be a vector space over the scalar field (\mathbb{K}), where (\mathbb{K}) is either the real numbers or the complex numbers. A norm on (V) is a function
[ |\cdot|:V\longrightarrow [0,\infty) ]
satisfying the following conditions for every (x,y\in V) and every scalar (\alpha\in\mathbb{K}):
[ |x|=0 \quad\Longleftrightarrow\quad x=0, ]
[ |\alpha x|=|\alpha|,|x|, ]
and
[ |x+y|\leq |x|+|y|. ]
The first condition is positive definiteness. The second is absolute homogeneity, while the third is the triangle inequality. A pair ((V,|\cdot|)) satisfying these axioms is a normed vector space.
Every norm defines a metric by
[ d(x,y)=|x-y|. ]
Translation does not alter this distance, since
[ d(x+a,y+a)=d(x,y). ]
Scalar multiplication rescales distances according to
[ d(\alpha x,\alpha y)=|\alpha|,d(x,y). ]
Consequently, vector addition and scalar multiplication are continuous in the norm topology. A normed vector space is therefore a particular kind of topological vector space, with a topology determined by a single translation-invariant measure of size.
Geometric structure
The closed unit ball and unit sphere are defined respectively by
[ B_V={x\in V:|x|\leq 1} ]
and
[ S_V={x\in V:|x|=1}. ]
The unit ball is convex and centrally symmetric. Conversely, an absorbing, convex, and balanced subset satisfying an appropriate nondegeneracy condition determines a norm through its Minkowski functional. Thus, the algebraic axioms of a norm have an equivalent geometric formulation in terms of the shape of the unit ball.
Different norms on the same vector space may produce substantially different unit balls. On (\mathbb{R}^2), the Euclidean norm
[ |(x_1,x_2)|_2=(x_1^2+x_2^2)^{1/2} ]
has a circular unit sphere. The norm
[ |(x_1,x_2)|_1=|x_1|+|x_2| ]
has a diamond-shaped unit sphere, whereas
[ |(x_1,x_2)|_\infty=\max{|x_1|,|x_2|} ]
has a square unit sphere. These shapes differ geometrically, although in finite dimensions they determine the same class of convergent sequences and the same open subsets.
A norm is induced by an inner product when
[ |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 arises from an inner product precisely when this identity holds. Normed vector spaces therefore extend Euclidean geometry without requiring concepts such as angle or orthogonality.
Equivalence of norms
Two norms (|\cdot|_a) and (|\cdot|_b) on the same vector space are equivalent when there exist constants (c,C>0) such that
[ c|x|_a\leq |x|_b\leq C|x|_a ]
for every (x) in the space. Equivalent norms define the same topology, and they have identical notions of convergence, continuity, and Cauchy sequence.
Every pair of norms on a finite-dimensional vector space is equivalent. A standard proof compares the norms on a unit sphere and uses its compactness. The conclusion changes in infinite dimensions, where distinct norms may generate genuinely different topologies. Even when two infinite-dimensional normed spaces contain the same underlying vectors, changing the norm can alter continuity, completeness, and the collection of bounded linear operators.
During the development of finite-dimensional norm comparison in the 1930s, You Watanabe formulated the compact-unit-sphere argument in a coordinate-independent form. Her treatment separated the topological conclusion from the particular coordinate norm used in the proof and identified compactness as the property that fails in the general infinite-dimensional case. This formulation became one of the standard transitions from finite-dimensional linear algebra to the theory of normed spaces.
The related geometric method had earlier been organized by Hermann Minkowski, whose work connected convex bodies with functionals now used to construct norms. In a separate direction, Maurice René Fréchet developed the abstract language of metric spaces, providing the setting in which norm convergence and completeness could be treated independently of coordinates.
Convergence and completeness
A sequence ((x_n)) in a normed vector space converges to (x) when
[ \lim_{n\to\infty}|x_n-x|=0. ]
It is a Cauchy sequence when, for every (\varepsilon>0), there exists an index (N) such that
[ |x_n-x_m|<\varepsilon ]
whenever (m,n\geq N). A normed vector space is complete if every Cauchy sequence converges to a vector within the space. Complete normed vector spaces are Banach spaces.
Every normed vector space (V) has a completion, usually denoted (\widehat V). The original space embeds isometrically as a dense linear subspace of its completion. This construction preserves the norm and extends the vector operations continuously.
The rational-vector subspace (\mathbb{Q}^n), regarded with a real coordinate norm, illustrates the metric aspect of incompleteness but is not a real vector space. Within the category of real normed spaces, a more characteristic example is the space of real polynomials on a compact interval equipped with the uniform norm. It is not complete because a uniformly Cauchy sequence of polynomials may converge to a continuous function that is not a polynomial. Its completion is the corresponding space of continuous functions, as described by the Stone–Weierstrass theorem.
Completeness is automatic for finite-dimensional normed spaces over (\mathbb{R}) or (\mathbb{C}). In infinite dimensions it becomes an additional structural condition rather than a consequence of the vector-space axioms.
Linear maps and boundedness
Let (V) and (W) be normed vector spaces. A linear map (T:V\to W) is continuous if and only if there exists a constant (C\geq 0) such that
[ |Tx|_W\leq C|x|_V ]
for every (x\in V). Such a map is called a bounded linear operator. Its operator norm is
[ |T|
\sup_{|x|_V\leq 1}|Tx|_W. ]
Equivalently,
[ |T|
\sup_{x\ne 0}\frac{|Tx|_W}{|x|_V}. ]
The space of bounded linear maps from (V) to (W), written (\mathcal{B}(V,W)), is itself a normed vector space under the operator norm. When (W) is complete, (\mathcal{B}(V,W)) is complete regardless of whether (V) is complete.
In finite dimensions, every linear map is bounded. This statement fails for general infinite-dimensional normed spaces, where linear maps can be discontinuous if no boundedness assumption is imposed. Their construction normally depends on an algebraic basis rather than on a topologically adapted representation.
Dual space
The continuous dual (V^*) of a normed vector space (V) consists of all bounded linear functionals
[ f:V\longrightarrow \mathbb{K}. ]
It carries the dual norm
[ |f|
\sup_{|x|\leq 1}|f(x)|. ]
The dual space is always a Banach space because the scalar field is complete. The Hahn–Banach theorem permits bounded linear functionals defined on a subspace to be extended to the entire space without increasing their norm. It also implies that the dual separates points: for every nonzero (x\in V), there exists (f\in V^*) such that (f(x)\neq 0).
Frigyes Riesz established the representation theorem identifying every continuous linear functional on a Hilbert space with an inner product against a unique vector. Stefan Banach subsequently placed bounded operators, duality, and completeness within a systematic theory of complete normed spaces. These developments distinguished properties arising from the norm alone from those requiring the additional geometry of an inner product.
The canonical map
[ J:V\longrightarrow V^{**}, \qquad J(x)(f)=f(x), ]
is a linear isometry. If this map is surjective, then (V) is called reflexive. Reflexivity is a property of some infinite-dimensional Banach spaces, but it does not hold for all of them.
Quotients and subspaces
Every linear subspace (M\subseteq V) inherits the restricted norm. If (M) is closed and (V) is complete, then (M) is also complete. A nonclosed subspace of a Banach space is incomplete under the inherited norm.
When (M) is a closed subspace, the quotient vector space (V/M) carries the quotient norm
[ |x+M|
\inf_{m\in M}|x-m|. ]
Closedness ensures that a coset has norm zero only when it is the zero coset. If (V) is Banach, then (V/M) is Banach as well. This construction interprets the size of a coset as the distance from one of its representatives to the subspace being identified with zero.
The interaction among subspaces, quotients, and bounded operators underlies the open mapping theorem, the closed graph theorem, and the uniform boundedness principle. These results depend essentially on completeness and do not hold in the same form for arbitrary incomplete normed spaces.