normed space

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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, a normed vector space or normed space is a
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
over the
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

numbers, on which a
norm Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy) Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...
is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" in the real world. A norm is a
real-valued function Mass measured in grams is a function from this collection of weight to positive number">positive Positive is a property of Positivity (disambiguation), positivity and may refer to: Mathematics and science * Converging lens or positive lens, i ...
defined on the vector space that is commonly denoted $x\mapsto \, x\, ,$ and has the following properties: #It is nonnegative, meaning that $\, x\, \geq 0$ for every vector $x.$ #It is positive on nonzero vectors, that is, $\, x\, = 0 \text x = 0.$ # For every vector $x,$ and every scalar $\alpha,$ $\, \alpha x\, = , \alpha, \, \, x\, .$ # The holds; that is, for every vectors $x$ and $y,$ $\, x+y\, \leq \, x\, + \, y\, .$ A norm induces a
distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...
, called its , by the formula $d(x,y) = \, y-x\, .$ which make any normed vector space into a
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
and a
topological vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. If this metric $d$ is complete then the normed space is a
Banach space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. Every normed vector space can be "uniquely extended" to a Banach space, which makes normed spaces intimately related to Banach spaces. Every Banach space is a normed space but converse is not true. For example, the set of the
finite sequence In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s of real numbers can be normed with the
Euclidean norm Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called pa ...
, but it is not complete for this norm. An
inner product space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
is a normed vector space whose norm is the square root of the inner product of a vector and itself. The
Euclidean norm Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called pa ...
of a
Euclidean vector space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called p ...
is a special case that allows defining
Euclidean distance In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
by the formula $d(A, B) = \, \overrightarrow\, .$ The study of normed spaces and Banach spaces is a fundamental part of
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
, which is a major subfield of mathematics.

Definition

A normed vector space is a
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
equipped with a
norm Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy) Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...
. A is a vector space equipped with a
seminorm In mathematics, particularly in functional analysis, a seminorm is a Norm (mathematics), vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some A ...
. A useful variation of the triangle inequality is $\, x-y\, \geq , \, x\, - \, y\, ,$ for any vectors $x$ and $y.$ This also shows that a vector norm is a
continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
. Property 3 depends on a choice of norm $, \alpha,$ on the field of scalars. When the scalar field is $\R$ (or more generally a subset of $\Complex$), this is usually taken to be the ordinary
absolute value In , the absolute value or modulus of a  , denoted , is the value of  without regard to its . Namely, if is , and if is (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of − ...

, but other choices are possible. For example, for a vector space over $\Q$ one could take $, \alpha,$ to be the $p$-adic norm.

Topological structure

If $\left(V, \, \,\cdot\,\, \right)$ is a normed vector space, the norm $\, \,\cdot\,\,$ induces a
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
(a notion of ''distance'') and therefore a
topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

on $V.$ This metric is defined in the natural way: the distance between two vectors $\mathbf$ and $\mathbf$ is given by $\, \mathbf - \mathbf\, .$ This topology is precisely the weakest topology which makes $\, \,\cdot\,\,$ continuous and which is compatible with the linear structure of $V$ in the following sense: #The vector addition $\,+\, : V \times V \to V$ is jointly continuous with respect to this topology. This follows directly from the . #The scalar multiplication $\,\cdot\, : \mathbb \times V \to V,$ where $\mathbb$ is the underlying scalar field of $V,$ is jointly continuous. This follows from the triangle inequality and homogeneity of the norm. Similarly, for any seminormed vector space we can define the distance between two vectors $\mathbf$ and $\mathbf$ as $\, \mathbf - \mathbf\, .$ This turns the seminormed space into a
pseudometric space In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. In the same way as every normed space is a metric space, every seminormed space is a pseudometric space. Becaus ...
(notice this is weaker than a metric) and allows the definition of notions such as continuity and
convergence Convergence may refer to: Arts and media Literature *Convergence (book series), ''Convergence'' (book series), edited by Ruth Nanda Anshen *Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics: **A four-par ...

. To put it more abstractly every seminormed vector space is a
topological vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
and thus carries a
topological structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
which is induced by the semi-norm. Of special interest are complete normed spaces, which are known as . Every normed vector space $V$ sits as a dense subspace inside some Banach space; this Banach space is essentially uniquely defined by $V$ and is called the of $V.$ Two norms on the same vector space are called if they define the same
topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...
. On a finite-dimensional vector space, all norms are equivalent but this is not true for infinite dimensional vector spaces. All norms on a finite-dimensional vector space are equivalent from a topological viewpoint as they induce the same topology (although the resulting metric spaces need not be the same)., Theorem 1.3.6 And since any Euclidean space is complete, we can thus conclude that all finite-dimensional normed vector spaces are Banach spaces. A normed vector space $V$ is
locally compactIn mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighbourhood (mathematics), ne ...
if and only if the unit ball $B = \$ is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British N ...
, which is the case if and only if $V$ is finite-dimensional; this is a consequence of Riesz's lemma. (In fact, a more general result is true: a topological vector space is locally compact if and only if it is finite-dimensional. The point here is that we don't assume the topology comes from a norm.) The topology of a seminormed vector space has many nice properties. Given a neighbourhood system $\mathcal\left(0\right)$ around 0 we can construct all other neighbourhood systems as $\mathcal(x) = x + \mathcal(0) := \$ with $x + N := \.$ Moreover, there exists a
neighbourhood basisIn topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point is the collection of all Neighbourhood (mathematics), neighbourhoods of the point . Definit ...
for the origin consisting of absorbing and
convex set File:Convex polygon illustration2.svg, Illustration of a non-convex set. Illustrated by the above line segment whereby it changes from the black color to the red color. Exemplifying why this above set, illustrated in green, is non-convex. In geome ...

s. As this property is very useful in
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
, generalizations of normed vector spaces with this property are studied under the name
locally convex space In functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional a ...
s. A norm (or
seminorm In mathematics, particularly in functional analysis, a seminorm is a Norm (mathematics), vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some A ...
) $\, \cdot\,$ on a topological vector space $\left(X, \tau\right)$ is continuous if and only if the topology $\tau_$ that $\, \cdot\,$ induces on $X$ is Comparison of topologies, coarser than $\tau$ (meaning, $\tau_ \subseteq \tau$), which happens if and only if there exists some open ball $B$ in $\left(X, \, \cdot\, \right)$ (such as maybe $\$ for example) that is open in $\left(X, \tau\right)$ (said different, such that $B \in \tau$).

Normable spaces

A
topological vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
$\left(X, \tau\right)$ is called normable if there exists a norm $\, \cdot \,$ on $X$ such that the canonical metric $\left(x, y\right) \mapsto \, y-x\,$ induces the topology $\tau$ on $X.$ The following theorem is due to Andrey Kolmogorov, Kolmogorov: Kolmogorov's normability criterion: A Hausdorff topological vector space is normable if and only if there exists a convex, von Neumann bounded neighborhood of $0 \in X.$ A product of a family of normable spaces is normable if and only if only finitely many of the spaces are non-trivial (that is, $\neq \$). Furthermore, the quotient of a normable space $X$ by a closed vector subspace $C$ is normable, and if in addition $X$'s topology is given by a norm $\, \,\cdot,\,$ then the map $X/C \to \R$ given by $x + C \mapsto \inf_ \, x + c\,$ is a well defined norm on $X / C$ that induces the quotient topology on $X / C.$ If $X$ is a Hausdorff Locally convex topological vector space, locally convex
topological vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
then the following are equivalent: # $X$ is normable. # $X$ has a bounded neighborhood of the origin. # the strong dual space $X^_b$ of $X$ is normable. # the strong dual space $X^_b$ of $X$ is Metrizable topological vector space, metrizable. Furthermore, $X$ is finite dimensional if and only if $X^_$ is normable (here $X^_$ denotes $X^$ endowed with the weak-* topology). The topology $\tau$ of the Fréchet space $C^\left(K\right),$ as defined in the article on spaces of test functions and distributions, is defined by a countable family of norms but it is a normable space because there does not exist any norm $\, \cdot\,$ on $C^\left(K\right)$ such that the topology that this norm induces is equal to $\tau.$ Even if a metrizable topological vector space has a topology that is defined by a family of norms, then it may nevertheless still fail to be normable space (meaning that its topology can not be defined by any norm). An example of such a space is the Fréchet space $C^\left(K\right),$ whose definition can be found in the article on spaces of test functions and distributions, because its topology $\tau$ is defined by a countable family of norms but it is a normable space because there does not exist any norm $\, \cdot\,$ on $C^\left(K\right)$ such that the topology this norm induces is equal to $\tau.$ In fact, the topology of a Locally convex topological vector space, locally convex space $X$ can be a defined by a family of on $X$ if and only if there exists continuous norm on $X.$

Linear maps and dual spaces

The most important maps between two normed vector spaces are the Continuous function (topology), continuous Linear transformation, linear maps. Together with these maps, normed vector spaces form a Category theory, category. The norm is a continuous function on its vector space. All linear maps between finite dimensional vector spaces are also continuous. An ''isometry'' between two normed vector spaces is a linear map $f$ which preserves the norm (meaning $\, f\left(\mathbf\right)\, = \, \mathbf\,$ for all vectors $\mathbf$). Isometries are always continuous and injective. A surjective isometry between the normed vector spaces $V$ and $W$ is called an ''isometric isomorphism'', and $V$ and $W$ are called ''isometrically isomorphic''. Isometrically isomorphic normed vector spaces are identical for all practical purposes. When speaking of normed vector spaces, we augment the notion of dual space to take the norm into account. The dual $V^$ of a normed vector space $V$ is the space of all ''continuous'' linear maps from $V$ to the base field (the complexes or the reals) — such linear maps are called "functionals". The norm of a functional $\varphi$ is defined as the supremum of $, \varphi\left(\mathbf\right),$ where $\mathbf$ ranges over all unit vectors (that is, vectors of norm $1$) in $V.$ This turns $V^$ into a normed vector space. An important theorem about continuous linear functionals on normed vector spaces is the Hahn–Banach theorem.

Normed spaces as quotient spaces of seminormed spaces

The definition of many normed spaces (in particular,
Banach space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s) involves a seminorm defined on a vector space and then the normed space is defined as the Quotient space (linear algebra), quotient space by the subspace of elements of seminorm zero. For instance, with the Lp space, $L^p$ spaces, the function defined by $\, f\, _p = \left( \int , f(x), ^p \;dx \right)^$ is a seminorm on the vector space of all functions on which the Lebesgue integral on the right hand side is defined and finite. However, the seminorm is equal to zero for any function Support (mathematics), supported on a set of Lebesgue measure zero. These functions form a subspace which we "quotient out", making them equivalent to the zero function.

Finite product spaces

Given $n$ seminormed spaces $\left\left(X_i, q_i\right\right)$ with seminorms $q_i : X_i \to \R,$ denote the product space by $X := \prod_^n X_i$ where vector addition defined as $\left(x_1,\ldots,x_n\right) + \left(y_1,\ldots,y_n\right) := \left(x_1 + y_1, \ldots, x_n + y_n\right)$ and scalar multiplication defined as $\alpha \left(x_1,\ldots,x_n\right) := \left(\alpha x_1, \ldots, \alpha x_n\right).$ Define a new function $q : X \to \R$ by $q\left(x_1,\ldots,x_n\right) := \sum_^n q_i\left(x_i\right),$ which is a seminorm on $X.$ The function $q$ is a norm if and only if all $q_i$ are norms. More generally, for each real $p \geq 1$ the map $q : X \to \R$ defined by $q\left(x_1,\ldots,x_n\right) := \left(\sum_^n q_i\left(x_i\right)^p\right)^$ is a semi norm. For each $p$ this defines the same topological space. A straightforward argument involving elementary linear algebra shows that the only finite-dimensional seminormed spaces are those arising as the product space of a normed space and a space with trivial seminorm. Consequently, many of the more interesting examples and applications of seminormed spaces occur for infinite-dimensional vector spaces.

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Banach space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, normed vector spaces which are complete with respect to the metric induced by the norm * * Finsler manifold, where the length of each tangent vector is determined by a norm * Inner product space, normed vector spaces where the norm is given by an inner product * * Locally convex topological vector space – a vector space with a topology defined by convex open sets * Space (mathematics) – mathematical set with some added structure *

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