In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a normed vector space or normed space is a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over the
real or
complex numbers, on which a
norm is defined.
A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" in the real (physical) world. A norm is a
real-valued function
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.
Real-valued functions of a real variable (commonly called ''real f ...
defined on the vector space that is commonly denoted
and has the following properties:
#It is nonnegative, meaning that
for every vector
#It is positive on nonzero vectors, that is,
# For every vector
and every scalar
# The
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.
This statement permits the inclusion of degenerate triangles, but ...
holds; that is, for every vectors
and
A norm induces a
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
, called its , by the formula
which makes any normed vector space into a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
and a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
. If this metric space is
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
then the normed space is a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
. 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 sequences of real numbers can be normed with the
Euclidean norm
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, but it is not complete for this norm.
An
inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
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 geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
of a
Euclidean vector space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
is a special case that allows defining
Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore ...
by the formula
The study of normed spaces and Banach spaces is a fundamental part of
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 (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
, which is a major subfield of mathematics.
Definition
A normed vector space is a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
equipped with a
norm. A is a vector space equipped with a
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk a ...
.
A useful
variation of the triangle inequality is
for any vectors
and
This also shows that a vector norm is a (uniformly)
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
.
Property 3 depends on a choice of norm
on the field of scalars. When the scalar field is
(or more generally a subset of
), this is usually taken to be the ordinary
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
, but other choices are possible. For example, for a vector space over
one could take
to be the
-adic absolute value.
Topological structure
If
is a normed vector space, the norm
induces a
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathe ...
(a notion of ''distance'') and therefore a
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
on
This metric is defined in the natural way: the distance between two vectors
and
is given by
This topology is precisely the weakest topology which makes
continuous and which is compatible with the linear structure of
in the following sense:
#The vector addition
is jointly continuous with respect to this topology. This follows directly from the
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.
This statement permits the inclusion of degenerate triangles, but ...
.
#The scalar multiplication
where
is the underlying scalar field of
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
and
as
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 seminormed vector space is a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
and thus carries a
topological structure
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
which is induced by the semi-norm.
Of special interest are
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
normed spaces, which are known as .
Every normed vector space
sits as a dense subspace inside some Banach space; this Banach space is essentially uniquely defined by
and is called the of
Two norms on the same vector space are called if they define the same
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
. 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
is
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
if and only if the unit ball
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 Britis ...
, which is the case if and only if
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 around 0 we can construct all other neighbourhood systems as
with
Moreover, there exists a
neighbourhood basis for the origin consisting of
absorbing and
convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
s. As this property is very useful in
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 (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
, generalizations of normed vector spaces with this property are studied under the name
locally convex space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
s.
A norm (or
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk a ...
)
on a topological vector space
is continuous if and only if the topology
that
induces on
is
coarser than
(meaning,
), which happens if and only if there exists some open ball
in
(such as maybe
for example) that is open in
(said different, such that
).
Normable spaces
A
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
is called normable if there exists a norm
on
such that the canonical metric
induces the topology
on
The following theorem is due to
Kolmogorov
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
:
Kolmogorov's normability criterion: A Hausdorff topological vector space is normable if and only if there exists a convex,
von Neumann bounded
In functional analysis and related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be ''inflated'' to include the set.
A set that is not bounded is ...
neighborhood of
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,
). Furthermore, the quotient of a normable space
by a closed vector subspace
is normable, and if in addition
's topology is given by a norm
then the map
given by
is a well defined norm on
that induces the
quotient topology
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...
on
If
is a Hausdorff
locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
then the following are equivalent:
#
is normable.
#
has a bounded neighborhood of the origin.
# the
strong dual space of
is normable.
# the strong dual space
of
is
metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
.
Furthermore,
is finite dimensional if and only if
is normable (here
denotes
endowed with the
weak-* topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
).
The topology
of the
Fréchet space 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
on
such that the topology that this norm induces is equal to
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 whose definition can be found in the article on
spaces of test functions and distributions, because its topology
is defined by a countable family of norms but it is a normable space because there does not exist any norm
on
such that the topology this norm induces is equal to
In fact, the topology of a
locally convex space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
can be a defined by a family of on
if and only if there exists continuous norm on
Linear maps and dual spaces
The most important maps between two normed vector spaces are the
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
linear maps. Together with these maps, normed vector spaces form a
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
.
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
which preserves the norm (meaning
for all vectors
). Isometries are always continuous and
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
. A
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
isometry between the normed vector spaces
and
is called an ''isometric isomorphism'', and
and
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
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
to take the norm into account. The dual
of a normed vector space
is the space of all ''continuous'' linear maps from
to the base field (the complexes or the reals) — such linear maps are called "functionals". The norm of a functional
is defined as the
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...
of
where
ranges over all unit vectors (that is, vectors of norm
) in
This turns
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, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
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
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
In measure theory, a branch of mathematics, 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 wi ...
zero. These functions form a subspace which we "quotient out", making them equivalent to the zero function.
Finite product spaces
Given
seminormed spaces
with seminorms
denote the
product space by
where vector addition defined as
and scalar multiplication defined as
Define a new function
by
which is a seminorm on
The function
is a norm if and only if all
are norms.
More generally, for each real
the map
defined by
is a semi norm.
For each
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
*
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
, normed vector spaces which are complete with respect to the metric induced by the norm
*
*
Finsler manifold
In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold where a (possibly asymmetric) Minkowski functional is provided on each tangent space , that enables one to define the length of any smooth curve ...
, where the length of each tangent vector is determined by a norm
*
Inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
, normed vector spaces where the norm is given by an
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
*
*
Locally convex topological vector space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
– a vector space with a topology defined by convex open sets
*
Space (mathematics)
In mathematics, a space is a set (sometimes called a universe) with some added structure.
While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, ...
– mathematical set with some added structure
*
References
Bibliography
*
*
*
*
*
External links
*
{{DEFAULTSORT:Normed Vector Space