Norm (mathematics)
A standard (from the Latin norma " guideline " ) is in mathematics a mapping that assigns a mathematical object, such as a vector, a matrix, a sequence or a function, a number that is to describe in some way the size of the object. The concrete meaning of " size " depends on the observed object and the standard used, for example, a standard, the length of a vector, the largest singular value of a matrix, the variation of a sequence or the maximum of a function represent. A norm is symbolized by two vertical bars on the left and right of the object.
Formally, a standard is a mapping that assigns a non-negative real number to an element of a vector space over the real or complex numbers and has the three properties definiteness, absolute homogeneity and subadditivity. A standard can ( but need not ) be derived from a scalar product. If a vector space equipped with a norm, we obtain a normed space with important analytical properties, since every norm induced on a vector space is also a metric and thus a topology. Two mutually equivalent standards thereby induce the same topology, where all norms are equivalent in finite dimensional vector spaces.
Standards are studied in particular in linear algebra and functional analysis, but they play an important role in numerical mathematics.
- 2.1 Number norms 2.1.1 Amount Standard
- 2.2.1 maximum norm
- 2.2.2 Euclidean norm
- 2.2.3 sum norm
- 2.2.4 p- norms
- 2.3.1 Matrix Norms on vector norms
- 2.3.2 matrix norms on operator norms
- 2.3.3 matrix norms on singular values
- 3.1 Normed spaces
- 3.2 Normalized algebras
- 3.3 seminorms
- 3.4 Equivalence of norms
- 3.5 Dual standards
- 4.1 follow- standards 4.1.1 supremum
- 4.1.2 bv standard
- 4.1.3 ℓ p- norms
- 4.2.1 supremum
- 4.2.2 BV - norm
- 4.2.3 maximum norm
- 4.2.4 Hölder norms
- 4.2.5 Essential supremum
- 4.2.6 Lp- norms
- 4.2.7 Cm - norms
- 4.2.8 Sobolev norms
- 4.3.1 operator norm
- 4.3.2 Nuclear Standard
- 4.3.3 Hilbert-Schmidt norm
- 4.3.4 Shadow standards
- 5.1 Weighted standards
- 5.2 Quasi standards
- 5.3 Standards on moduli
Basic concepts
Definition
A norm is a mapping from a vector space over the field of real or complex numbers into the set of nonnegative real numbers,
For all vectors and all scalars, the following three properties ( axioms ) has:
- ( Definiteness )
- ( absolute homogeneity )
- ( Subadditivity or triangle )
This refers to the amount of the scalar.
This axiomatic definition of the standard was set up by Stefan Banach in 1922 in his dissertation. The now common standard symbol was first used by Erhard Schmidt in 1908 used as the distance between vectors and.
Example
The standard example of a norm is the Euclidean norm, the ( originating in the zero point ) corresponds to the descriptive length of a vector in the plane or in space. For example, the Euclidean norm of the vector according to the Pythagorean theorem is the same. The definiteness then means that, when the length of a vector is null, this must be the zero vector. The absolute homogeneity states that if each component of a vector is multiplied by a number, its length changes with the value of this number. Finally, the triangle indicates that the length of the sum of two vectors is at most as large as the sum of the two lengths.
Basic Properties
From the absolute homogeneity follows by setting directly
Is the reversed direction of definiteness. Therefore, a vector has exactly zero then the standard if it is the zero vector. Furthermore, it follows from the absolute homogeneity by setting
So symmetry with respect to sign reversal. From the triangle inequality then follows by setting that a standard is always non-negative, ie
Applies. Thus, each different from the zero vector vector has a positive norm. Furthermore, applies to standards, the reverse triangle inequality
Which can be demonstrated by applying the triangle inequality, and taking into account the symmetry. For each standard is a uniformly continuous map. In addition, a standard is due to the sub-additivity and absolute homogeneity of a sub-linear, and thus convex figure, that is valid for all
Standard balls
For a given vector and a scalar is, the amount
Open or closed norm ball and the amount
Standard sphere with radius around. The terms " ball " or "sphere" are very common to see - for example, a standard ball into corners and edges have - and coincide only in the special case of the Euclidean vector norm with the ball concept known from the geometry. If we choose in the definition and so is called the resulting quantities unit sphere or unit sphere. Each standard or norm ball sphere arises from the corresponding unit ball or the unit sphere by scaling by the factor and translation by the vector. A vector of the unit sphere is called a unit vector; to each vector is obtained by normalizing the corresponding unit vector.
In any case, a standard sphere is a convex set, since otherwise the corresponding figure would not satisfy the triangle inequality. Furthermore, a standard ball due to the absolute homogeneity must always be point- symmetrical with respect. A norm can be defined in finite dimensional vector spaces using the associated standard ball, if this amount is convex, point-symmetric with respect to the zero point, is closed and bounded and has the zero point in the interior. The corresponding figure is also called Minkowski functional or functional verification. Hermann Minkowski examined such calibration Functional in 1896 under number-theoretic questions.
Induced norms
A standard can be, but need not necessarily be derived from a scalar product. The norm of a vector is then defined as
So the square root of the scalar product of the vector with itself is called in this case induced by the scalar product norm or Hilbert norm. Each induced by an inner product norm satisfies the Cauchy- Schwarz inequality
And is invariant under unitary transformations. By the theorem of Jordan -von Neumann a norm is induced by an inner product if and only if it satisfies the parallelogram law. Some important standards are not derived from a scalar product; historically existed even an essential step in the development of functional analysis in the introduction of standards that are not based on a scalar. However, there is an associated semi- inner product for each standard.
Norms on finite dimensional vector spaces
Number of standards
Standard amount
The amount of a real number is a simple example of a norm. The standard amount is obtained by omitting the sign of the number, so
The magnitude of a complex number is in accordance with this
Defined, wherein the complex conjugate number to, and respectively specifying the real and imaginary parts of the complex number. The magnitude of a complex number corresponds to the length of its vector in the Gaussian plane.
The amount of the standard is two standard scalar product of real or complex numbers
Induced.
Vector norms
The following real or complex vectors of finite dimension are considered. A vector (in the narrow sense) is then a tuple with entries for. For the following definitions, it is irrelevant whether it is a row or a column vector. For all comply with the following standards or standard amount of the previous section.
Maximum norm
The maximum norm, Chebyshev norm or ∞ - norm ( infinity - norm ) of a vector is defined as
And corresponds to the amount of the amount the largest component of the vector. Has the unit sphere of the real maximum norm in two dimensions the shape of a square, in three dimensions, the shape of a cube and dimensions in general the form of a hypercube.
The maximum norm is not induced by a scalar product. The means of their derived metric maximum metric, Chebyshev metric or, in particular, in two dimensions, chessboard metric because it measures the distance corresponding to the number of steps that must make a king in chess order from a field on the chessboard to come to a different field. Since the king can drag diagonally, for example, the distance between the centers of the two diagonally opposite corner squares of a chessboard in the maximum metric equal.
The maximum norm is a special case of the product standard
With over the product space of normed vector spaces and.
Euclidean norm
The Euclidean norm, or 2- norm of a vector is defined as
And complies with the square root of the sum of the absolute squares of the components of the vector. For real vectors, however, can be omitted in the definition of the modulus signs in complex vectors not.
The unit sphere of the real Euclidean norm in two dimensions has the shape of a circle, in three dimensions the shape of a spherical surface and in the general dimensions of the shape of a sphere. In two and three dimensions, the Euclidean norm of a vector describing the illustrative length in the plane or in space. The Euclidean norm is the only vector norm is invariant under unitary transformations, such as rotations of the vector to zero.
The Euclidean norm is given by the standard scalar product of two real or complex vectors by
Induced. A provided with the Euclidean norm vector space is called a Euclidean space. The derived from the Euclidean norm metric is called Euclidean metric. For example, the distance between the centers of the two diagonally opposite corner squares of a chessboard in the Euclidean metric, by the theorem of Pythagoras is the same.
Sum norm
The sum norm (exact ) Total amount norm or 1- norm (read: "one standard" ) of a vector is defined as
And the sum of the amounts of the components of the vector. The unit sphere of the real sum norm is in two dimensions, the shape of a square, in three dimensions the shape of an octahedron and the general dimensions of the shape of a Kreuzpolytops.
The sum norm is not induced by a scalar product. The derived from the sum norm metric is, especially in the real two-dimensional space also Manhattan metric or taxi metric because it measures the distance between two points as the route on a grid-like map, where you can only move in vertical and horizontal sections. For example, the distance between the centers of the two diagonally opposite corner squares of a chessboard in the Manhattan metric is the same.
P- norms
General it can be for real, the p- norm of a vector by
Define. For one obtains the sum norm, for the Euclidean norm and as a limit for the maximum norm. The unit spheres of the p- norms in the real case in two dimensions in the form of super- ellipses or Subellipsen and in three and higher dimensions, the shape of Superellipsoiden or Subellipsoiden.
All p- norms, including the maximum norm satisfy the Minkowski inequality and the Hölder 's inequality. You are falling for growing monotonous and equivalent to each other. As Narrowing factors arise for
Where in the case of the maximum norm of the exponent is set. The p- norms thus differ by a maximum factor. The images of the p- standards are not defined analogue standards, since the resulting standard balls are not convex, and thus the triangle inequality is violated.
Matrix norms
Hereinafter real or complex matrices are considered with rows and columns. For matrix norms, in addition to the three standard properties sometimes the Submultiplikativität
Requires defining with more than property. Is a matrix norm submultiplicative, the spectral matrix (the sum of the absolute value largest eigenvalue ) is at most as large as the norm of the matrix. However, there are matrix norms with the usual standard properties which are not submultiplicative. Most in the definition of a matrix norm is a vector norm is assumed. A matrix norm is this compatible with a vector norm if
Applies to all.
Matrix norms on vector norms
By all entries of a matrix are written below each other, a matrix can be viewed as a vector of appropriate length. This matrix norms can be defined directly via vector norms, in particular the p- norms by
Where the entries of the matrix. Examples of such matrix defined standards are based on the maximum-norm total norm and the Frobenius norm based on the Euclidean norm, both of which are compatible with submultiplicative and the Euclidean norm.
Matrix norms on operator norms
A matrix norm is a vector norm induced or natural matrix norm if it is derived as an operator norm, ie if the following applies:
Clearly there corresponds a matrix norm defined as the maximum stretch factor on the application of the matrix to a vector. The operator norms such matrix norms are submultiplicative and always with the vector norm from which they were derived tolerated. An operator norm is even among all compatible with a vector norm matrix norms one with the smallest value. Examples of such matrix defined standards are based on the maximum norm row sum norm, which is based on the Euclidean norm and spectral norm which is based on the sum of standard column sum norm.
Matrix norms on singular values
Another way to derive matrix norms on vector norms, it is a singular value decomposition of a matrix into a unitary matrix, a diagonal matrix and a unitary matrix adjoint to look at. The nonnegative real entries are then the singular values of and equal to the square roots of the eigenvalues of. The singular values are then combined into a vector whose vector norm is considered, ie
Examples of thus defined matrix norms are defined on the p- norms of the vector of singular values shadows standards and based on the sum of the largest singular values of Ky Fan norms.
Further terms
Normed spaces
If a vector space equipped with a norm, we obtain a normed space with important analytical properties. For each induced norm between vectors by subtraction a metric
This Fréchet metric is a normed space to a metric space and continue with the induced from the metric topology is a topological space, even to a Hausdorff space. The standard is then a continuous map with respect to this norm topology. A series aims has been to a limit if and only if the following holds. Converges in a normed space every Cauchy sequence to a limit in this space, it is called a complete normed space or Banach space.
Normalized algebras
Provides you the vector space also with an associative and distributive vector product, then is an associative algebra. Is now a normed space and this standard submultiplicative, that is valid for all vectors
Then we obtain a normed algebra. If the normed space completely, one also speaks of a Banach algebra. For example, the space of square matrices with the matrix addition and multiplication, as well as a submultiplikativen matrix norm is such a Banach algebra.
Seminorms
If omitted, the first standard axiom definiteness, then is only a semi-norm (respectively a seminorm ). Due to the homogeneity and the subadditivity then the amount
Of vectors with norm zero is a subspace of. In this manner, an equivalence relation on by
Be defined. Is now identified in a new space so all equivalent elements as the same, then along with the norm is a normed space. This process is called residue class education in relation to the semi-norm and called the factor space. Through a set of semi-norms can also be special topological vector spaces, the locally convex spaces, define.
Equivalence of norms
Two standards and are called equivalent if there exist positive constants and such that for all
Applies, so if a standard can be estimated by other standard according to the up and down. Equivalent standards induce the same topology. A sequence converges with respect to a standard, it also converges with respect to an amount equivalent to their standard.
On finite-dimensional vector spaces, all norms are equivalent, since the standard balls are then compact sets. In infinite-dimensional spaces, however, not all norms are equivalent. Is a vector space but with respect to two standards completely, these two standards are already then equivalent if there is a positive constant, so that
Is true because there exists a continuous linear mapping between the two Banach spaces whose inverse according to the principle of continuous inverse is also continuous.
Dual standards
The dual space of a normed vector space over a field is the space of continuous linear functionals from to. For example, the dual chamber can the space of the n-dimensional (column) vectors are regarded as the space of the linear combination of the vector components, ie the area of the row vectors of the same dimension. The dual to a standard norm of functionals is then defined by
With this standard, the dual space is also a normed space. The dual space with the dual norm is always complete, regardless of the completeness of the output space. If two norms are equivalent, then the corresponding dual norms are also equivalent to each other. For dual standards is clear from the above definition as supremum immediately following important inequality
Standards on infinite-dimensional vector spaces
Consequence standards
Now, real - or complex-valued sequences with sequence elements for are considered. Consequences are thus a direct generalization of vectors of finite dimension. In contrast to finite-dimensional vectors consequences can be unlimited, making the previous vector norms can not be transferred directly to consequences. For example, the maximum amount or the amount sum of terms of the sequence of an unlimited sequence is infinite and thus not a real number more. Therefore, the observed sequence spaces must be restricted accordingly, so that the associated norms are finite.
Supremum
The supremum of a bounded sequence is defined as
The set of bounded sequences, the set of convergent sequences and the set of zero convergent sequences ( zero sequences ), together with the supremum norm complete normed spaces.
Bv- Standard
The bv- norm of a sequence of bounded variation is defined as
With the bv- norm of the sequence space is a complete normed space, since every sequence of bounded variation is a Cauchy sequence. For the sub- space of null sequences of bounded variation BV0 the standard is obtained by omitting the first term, ie
And with this standard, the room is complete.
ℓ p- norms
The ℓ p- norms are the generalization of the p- norms on sequence spaces, with only the finite sum is replaced by an infinite. The ℓ p- norm of a magnitude as in the p- th power summable sequence is defined for real then as
Provided with these standards, the ℓ p- spaces each to complete normed spaces. For the limit results in the space of bounded sequences with the supremum norm. The space is a Hilbert space with the scalar product
Two sequences. The ℓ p-norm with a dual norm with the ℓ q- norm. The space is not dual to the space, but dual to the space of convergent sequences and the space of null sequences each with the supremum norm.
Function standards
The following real - or complex-valued functions are considered on a lot. Often is a topological space, so you can talk about continuity, in many applications is a subset of the. As consequences, functions can in principle be unlimited. Therefore, the considered function spaces must be restricted accordingly, so that the associated norms are finite. The most important of these function spaces are classes limited, continuous, integrable and differentiable functions. More generally, the following function spaces and norms can also be defined for Banach space - valued functions, if the absolute value is replaced by the norm of the Banach space.
Supremum
The supremum of a bounded function, ie a function whose image is a bounded subset of is defined as
The set of bounded functions with the supremum norm is a complete normed space.
BV - norm
The BV - norm of a one-dimensional function with bounded variation on an interval is defined in analogy to the bv- norm of a sequence as
With a partition of the interval and the supremum is taken over all possible partitions. A function is of bounded variation if and only if it can be represented as a sum of a monotonically increasing and a monotonically decreasing function. The set of functions bounded variation with the BV - norm is a complete normed space. Alternatively, may be chosen as normalization term also take the integral of the function over the interval. For BV - norms and the corresponding spaces of functions of bounded variation, there are a number of multi-dimensional generalizations, such as the Fréchet variation, the Vitali variation and the Hardy variation.
Maximum norm
The maximum norm of a continuous function on a compact set is defined as
After the extreme value theorem assumes a continuous function on a compact set to its maximum. The space of continuous functions on a compact set is equipped with the maximum norm a complete normed space.
Hölder norms
The Hölder norm of a Hölder continuous function with Hölderexponent is defined as
The Hölder constant of the function by
Is given. The Hölder constant is a special form of continuity module and provides even half standard represents the spaces of Hölder continuous functions are with the respective Hölder norms complete normed spaces. In the special case one speaks of a Lipschitz continuous function, the Lipschitz constant and the Lipschitz norm.
Essential supremum
The standard almost everywhere bounded function on a measure space is defined as
Being a null set, ie an element of the algebra with measure zero, is. An almost everywhere bounded function can therefore assume at some point a magnitude higher value than its essential supremum. The essential supremum is generally only a half- standard, since the set of functions with zero standard includes not only the zero function, but also for example all functions that accept them differently on null sets non-zero values . Therefore, considering the set of equivalence classes of functions which are the same everywhere, and calls the corresponding factor space. In this space the essential supremum norm is defined as
In fact a standard, where the value on the right side is independent of the choice of the representatives of the equivalence class. It is often inaccurately written instead, in which case assumed that only a representative of the equivalence class. The space of equivalence classes of essentially bounded functions with the essential supremum norm is a complete normed space.
Lp- norms
The standards in a p- th power Lebesgue integrable function with are defined in analogy to the ℓ p- norms as
Wherein the sum has been replaced by an integral. As with the essential supremum norm these standards are initially only half standards because not only the zero function, but also all the functions that differ only on a set of measure zero of the zero function are integrated to zero. Therefore, we consider again the set of equivalence classes of functions which are the same everywhere, and defines this space Lp Lp norms by
By the theorem of Fischer- Riesz all Lp- spaces with the respective Lp norm are complete normed spaces. The space is the space of ( equivalence classes of ) Lebesgue integrable functions. The space of square integrable functions is a Hilbert space with scalar product
And for the limit results in the space of essentially bounded functions. The to the Lp norm for dual norm is with the Lq norm. The Lp- norms and spaces can be generalized by the Lebesgue measure on general measures, in which the duality only in certain measure spaces holds, see duality of Lp- spaces.
Cm - norms
The Cm - norm of a m- times continuously differentiable function on an open set whose partial derivatives are continuous continuable on the closure of the set is defined as
With a multi- index of non-negative integers, the associated mixed partial derivative of the function and the order of the derivative are. The C0 - norm corresponds to the supremum norm and the C1 norm to the maximum of the function and its first derivatives. The rooms are complete with the respective Cm norm normed spaces. Alternatively, the Cm - norm is defined by the sum of the individual standards rather than maximum, but both norms are equivalent.
Analogously, the Cm, α - norm of a m- times continuously differentiable function on an open set whose mixed partial derivatives are continuous continuable on the closure of the set and their Hölder constants of the derivatives are limited to degree, defined as
The rooms of this Hölder continuously differentiable functions with the respective Cm, α - norms also complete normed spaces.
Sobolev norms
The Sobolev norm of a m- times weakly differentiable function on an open set whose mixed weak derivatives up to the degree in p- th power Lebesgue integrable, for defined as
And for as
Looking in the sum only the mixed derivatives of order, so you only get half standard that is less than vanishes on all polynomials of degree. The Sobolev spaces of functions whose weak derivatives are mixed to the degree in, are full with the respective Sobolev norm normed spaces. In particular, the spaces are Hilbert spaces with scalar product
Sobolev norms play an important role in the solution theory of partial differential equations as natural domains of definition of the differential operators or error estimates of finite element method for the discretization of partial differential equations.
Standards on operators
The following are linear operators between two vector spaces and considered. It is assumed that these vector spaces are already self- normed spaces.
Operator norm
The operator norm of a bounded linear operator between two normed spaces is defined as
Is a linear map between finite dimensional vector spaces, its operator norm is a natural matrix norm by choosing a base. If the vector space completely, then the space of bounded (and therefore continuous ) linear operators from complete after. Operator norms are always submultiplicative, are therefore the two vector spaces equally and fully, then the space of continuous linear operators with the operator norm and the composition is a Banach algebra.
Nuclear norm
The nuclear norm of a nuclear operator between two Banach spaces is defined as
Wherein a sequence of vectors in the dual space, and a series of vectors, such that the shape has, and the infimum is taken over all such nuclear representations. If the two vector spaces Hilbert spaces is also called trace norm the corresponding nuclear norm. The space of nuclear operators is with the nuclear norm is a complete normed space.
Hilbert-Schmidt norm
The Hilbert-Schmidt norm of a Hilbert-Schmidt operator between two Hilbert spaces is defined as
Being an orthonormal basis of is. The Hilbert-Schmidt norm generalized to the case of the Frobenius norm of infinite Hilbert spaces. The Hilbert-Schmidt norm induced by the inner product, the adjoint operator is closed. The set of Hilbert - Schmidt operators forms with the Hilbert-Schmidt norm itself a Hilbert space and a Banach algebra, even an H * - algebra.
Shadow standards
The shadow - p-norm of a compact linear operator between two separable Hilbert spaces is defined as
The sequence of the singular values of the operator. In the event, the trace norm and in the case of the Hilbert-Schmidt norm. The set of compact linear operators whose singular values lie in, together with the respective shadow - p-norm a complete normed space is a Banach algebra and for.
Generalizations
Weighted norms
Weighted norms are norms on weighted vector spaces. For example, we obtain induced weighted function Enormous over by multiplication with a suitable positive weight function
With a weighted L2 - scalar product. The introduction of weight functions allows to extend function rooms, for example, functions for which the norm in the unweighted case would be unlimited, or to restrict, for example, to functions that have a certain drop behavior.
Quasi standards
If the triangle inequality to the effect weakened that only exists a real constant such that for all
Applies, it is called the corresponding figure quasi standard and provided with such a quasi- standard vector space quasi- normed space. For example, the ℓ p- norms for quasi norms and the associated ℓ p- spaces quasi- normed spaces, even quasi- Banach spaces.
Standards on moduli
The notion of a norm can be taken much more general, by the vector space is replaced by an R (left ) module over a unitary ring with amount. A function is then called to the standard module if for all and all scalars are met the three standard properties of definiteness, absolute homogeneity and subadditivity. If the amount is replaced by a pseudo amount in the base ring and the module homogeneity is attenuated to Subhomogenität, one obtains a pseudo- norm.