Normed vector space
Encyclopedia
In mathematics
, with 2- or 3-dimensional vectors with real
-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space
Rn. The following properties of "vector length" are crucial.
1. The zero vector, 0, has zero length; every other vector has a positive length.
if 
2. Multiplying a vector by a positive number changes its length without changing its direction. Moreover,
for any scalar 
3. The triangle inequality
holds. That is, taking norms as distances, the distance from point A through B to C is never shorter than going directly from A to C, or the shortest distance between any two points is a straight line.
for any vectors x and y. (triangle inequality)
The generalization of these three properties to more abstract vector space
s leads to the notion of norm
. A vector space on which a norm is defined is then called a normed vector space.
Normed vector spaces are central to the study of linear algebra and functional analysis.
and p a seminorm
on V.
A normed vector space is a pair (V, ‖·‖ ) where V is a vector space and ‖·‖ a norm on V.
We often omit p or ‖·‖ and just write V for a space if it is clear from the context what (semi) norm we are using.
In a more general sense, a vector norm can be taken to be any real-valued function that satisfies these three properties. The properties 1. and 2. together imply that
if and only if 
.
A useful variation of the triangle inequality is
for any vectors x and y.
This also shows that a vector norm is a continuous function
.
(a notion of distance) and therefore a topology
on V. This metric is defined in the natural way: the distance between two vectors u and v is given by ‖u−v‖. This topology is precisely the weakest topology which makes ‖·‖ continuous and which is compatible with the linear structure of V in the following sense:
Similarly, for any semi-normed vector space we can define the distance between two vectors u and v as ‖u−v‖. This turns the seminormed space into a pseudometric space
(notice this is weaker than a metric) and allows the definition of notions such as continuity and convergence
.
To put it more abstractly every semi-normed vector space is a topological vector space
and thus carries a topological structure which is induced by the semi-norm.
Of special interest are complete
normed spaces called Banach space
s. Every normed vector space V sits as a dense subspace inside a Banach space; this Banach space is essentially uniquely defined by V and is called the completion of V.
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). 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 compact if and only if the unit ball B = {x : ‖x‖ ≤ 1} is compact
, 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 has many nice properties. Given a neighbourhood system
around 0 we can construct all other neighbourhood systems as
with
.
Moreover there exists a neighbourhood basis for 0 consisting of absorbing
and convex set
s. As this property is very useful in functional analysis
, generalizations of normed vector spaces with this property are studied under the name locally convex spaces.
. Together with these maps, normed vector spaces form a 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(v)‖ = ‖v‖ for all vectors v). 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 φ is defined as the supremum
of |φ(v)| where v ranges over all unit vectors (i.e. 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
.
s) involves a seminorm defined on a vector space and then the normed space is defined as the quotient space
by the subspace of elements of seminorm zero. For instance, with the Lp spaces
, the function defined by
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 supported
on a set of Lebesgue measure
zero. These functions form a subspace which we "quotient out", making them equivalent to the zero function.
with vector addition defined as
and scalar multiplication defined as
.
We define a new function q
for example as
.
which is a seminorm on X. The function q is a norm if and only if all qi are norms.
More generally, for each real p≥1 we have the seminorm:
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, with 2- or 3-dimensional vectors with real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
Rn. The following properties of "vector length" are crucial.
1. The zero vector, 0, has zero length; every other vector has a positive length.
if 2. Multiplying a vector by a positive number changes its length without changing its direction. Moreover,
for any scalar 3. The triangle inequality
Triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side ....
holds. That is, taking norms as distances, the distance from point A through B to C is never shorter than going directly from A to C, or the shortest distance between any two points is a straight line.
for any vectors x and y. (triangle inequality)The generalization of these three properties to more abstract vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s leads to the notion of norm
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
. A vector space on which a norm is defined is then called a normed vector space.
Normed vector spaces are central to the study of linear algebra and functional analysis.
Definition
A seminormed vector space is a pair (V,p) where V is a vector spaceVector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
and p a seminorm
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
on V.
A normed vector space is a pair (V, ‖·‖ ) where V is a vector space and ‖·‖ a norm on V.
We often omit p or ‖·‖ and just write V for a space if it is clear from the context what (semi) norm we are using.
In a more general sense, a vector norm can be taken to be any real-valued function that satisfies these three properties. The properties 1. and 2. together imply that
if and only if .A useful variation of the triangle inequality is
for any vectors x and y.This also shows that a vector norm is a continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
.
Topological structure
If (V, ‖·‖) is a normed vector space, the norm ‖·‖ induces a metricMetric (mathematics)
In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
(a notion of distance) and therefore a topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
on V. This metric is defined in the natural way: the distance between two vectors u and v is given by ‖u−v‖. This topology is precisely the weakest topology which makes ‖·‖ continuous and which is compatible with the linear structure of V in the following sense:
- The vector addition + : V × V → V is jointly continuous with respect to this topology. This follows directly from the triangle inequalityTriangle inequalityIn mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side ....
. - The scalar multiplication · : K × V → V, where K is the underlying scalar field of V, is jointly continuous. This follows from the triangle inequality and homogeneity of the norm.
Similarly, for any semi-normed vector space we can define the distance between two vectors u and v as ‖u−v‖. This turns the seminormed space into a pseudometric space
Pseudometric space
In mathematics, a pseudometric space is a generalized 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...
(notice this is weaker than a metric) and allows the definition of notions such as continuity and convergence
Limit of a function
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....
.
To put it more abstractly every semi-normed vector space is a topological vector space
Topological vector space
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
and thus carries a topological structure which is induced by the semi-norm.
Of special interest are complete
Complete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
normed spaces called Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s. Every normed vector space V sits as a dense subspace inside a Banach space; this Banach space is essentially uniquely defined by V and is called the completion of V.
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). 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 compact if and only if the unit ball B = {x : ‖x‖ ≤ 1} is compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
, which is the case if and only if V is finite-dimensional; this is a consequence of Riesz's lemma
Riesz's lemma
Riesz's lemma is a lemma in functional analysis. It specifies conditions which guarantee that a subspace in a normed linear space is dense.- The result :...
. (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 has many nice properties. Given a neighbourhood system
around 0 we can construct all other neighbourhood systems aswith
.Moreover there exists a neighbourhood basis for 0 consisting of absorbing
Absorbing set
In functional analysis and related areas of mathematics an absorbing set in a vector space is a set S which can be inflated to include any element of the vector space...
and convex set
Convex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
s. As this property is very useful in functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, generalizations of normed vector spaces with this property are studied under the name locally convex spaces.
Linear maps and dual spaces
The most important maps between two normed vector spaces are the continuous linear mapsLinear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
. Together with these maps, normed vector spaces form a category
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
.
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(v)‖ = ‖v‖ for all vectors v). 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
Dual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
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 φ is defined as the supremum
Supremum
In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...
of |φ(v)| where v ranges over all unit vectors (i.e. 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
Hahn–Banach theorem
In mathematics, the Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed...
.
Normed spaces as quotient spaces of seminormed spaces
The definition of many normed spaces (in particular, Banach spaceBanach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s) involves a seminorm defined on a vector space and then the normed space is defined as the quotient space
Quotient space (linear algebra)
In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N ....
by the subspace of elements of seminorm zero. For instance, with the Lp spaces
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
, the function defined by
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 supported
Support (mathematics)
In mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set . This concept is used very widely in mathematical analysis...
on a set of Lebesgue measure
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
zero. These functions form a subspace which we "quotient out", making them equivalent to the zero function.
Finite product spaces
Given n seminormed spaces Xi with seminorms qi we can define the product space aswith vector addition defined as
and scalar multiplication defined as
.We define a new function q
for example as
.which is a seminorm on X. The function q is a norm if and only if all qi are norms.
More generally, for each real p≥1 we have the seminorm:
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.
See also
- locally convex spaces, generalizations of seminormed vector spaces
- Banach spaces, normed vector spaces which are complete with respect to the metric induced by the norm
- inner product spaces, normed vector spaces where the norm is given by an inner product
- Finsler manifoldFinsler manifoldIn mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold together with the structure of an intrinsic quasimetric space in which the length of any rectifiable curve is given by the length functional...
- Space (mathematics)