Norm (mathematics)
In mathematics, a norm (from the Latin norma "guideline") is a mapping that assigns a number to a mathematical object , for example a vector , a matrix , a sequence or a function , which is supposed to describe the size of the object in a certain way . The specific meaning of “size” depends on the object under consideration and the norm used, for example a norm can represent the length of a vector, the largest singular value of a matrix, the variation of a sequence or the maximum of a function. A standard is symbolized by two vertical lines to the left and right of the object.
Formally, a norm 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 of definiteness , absolute homogeneity and subadditivity . A norm can (but does not have to) be derived from a scalar product . If a vector space is provided with a norm, a normalized space with important analytical properties is obtained, since every norm induces a metric and thus a topology on a vector space . Two norms that are equivalent to one another induce the same topology, with all norms being equivalent to one another in finite-dimensional vector spaces.
Norms are studied in particular in linear algebra and functional analysis , but they also play an important role in numerical mathematics .
Basic concepts
definition
A norm is a mapping of a vector space over the field of real or complex numbers into the set of nonnegative real numbers ,
- ,
which satisfies the following three axioms for all vectors and all scalars :
(1) Definiteness : | , |
(2) absolute homogeneity : | , |
(3) Subadditivity or triangle inequality : | . |
Here denotes the amount of the scalar.
This axiomatic definition of the norm was established by Stefan Banach in his dissertation in 1922. The standard symbol common today was first used by Erhard Schmidt in 1908 as the distance between vectors and .
example
The standard example of a norm is the Euclidean norm of a vector (originating at the origin ) in the plane ,
- ,
which corresponds to the illustrative length of the vector. For example, the Euclidean norm of the vector is the same . Definiteness then means that if the length of a vector is zero , it must be the zero vector . Absolute homogeneity says that when each component of a vector is multiplied by a number, its length changes by a factor of the magnitude of that number. The triangle inequality finally states 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 it follows by setting directly
- ,
thus the opposite direction of definiteness. Therefore a vector has the norm zero if and only if it is the zero vector. Furthermore, it follows from the absolute homogeneity by setting
- and thus ,
thus symmetry with regard to sign reversal . By setting of the triangle inequality, it follows that a norm is always nonnegative , i.e.
applies. Thus every vector different from the zero vector has a positive norm. The reverse triangle inequality also applies to standards
- ,
which can be shown by applying the triangle inequality to and considering symmetry. Every norm is thus a uniformly continuous mapping . In addition, a standard result of the absolute homogeneity subadditivity and a sublinear and convex figure , that is, for all applicable
- .
Standard balls
For a given vector and a scalar with the set is called
- or.
open or closed standard sphere and the amount
Norm sphere around with radius . The terms " sphere " or " sphere " are to be seen very generally - for example, a standard sphere can also have corners and edges - and only in the special case of the Euclidean vector standard coincide with the concept of sphere known from geometry . One chooses in the definition and so is called the amounts resulting unit ball and unit sphere . Each standard sphere or sphere arises from the corresponding unit sphere or unit sphere by scaling with the factor and translation around the vector . A vector of the unit sphere is called a unit vector ; the associated unit vector is obtained for each vector by normalization .
In any case, a norm sphere must be a convex set , otherwise the corresponding mapping would not satisfy the triangle inequality. Furthermore, due to its absolute homogeneity, a standard sphere must always be point-symmetrical with respect to it. A norm can also be defined in finite-dimensional vector spaces via the associated norm sphere, if this set is convex, point-symmetrical with respect to the zero point, closed and limited and has the zero point inside . The corresponding mapping is also called a Minkowski functional or gauge functional. Hermann Minkowski examined such calibration functionals as early as 1896 in the context of number theoretic problems.
Induced norms
A norm can, but does not necessarily have to be, derived from a scalar product . The norm of a vector is then defined as
- ,
thus the root of the scalar product of the vector with itself. In this case one speaks of the norm induced by the scalar product or Hilbert norm. Every norm induced by a scalar product satisfies the Cauchy-Schwarz inequality
and is invariant under unitary transformations . According to Jordan-von Neumann's theorem , a norm is induced by a scalar product if and only if it satisfies the parallelogram equation. However, some important norms are not derived from an inner product; Historically, an essential step in the development of functional analysis was the introduction of standards that are not based on a scalar product. However, there is an associated semi-internal product for each standard .
Norms on finite-dimensional vector spaces
Number norms
Amount norm
The absolute value of a real number is a simple example of a norm. The amount norm is obtained by omitting the sign of the number, i.e.
The amount of a complex number is correspondingly through
defined, wherein the complex conjugate number to and respectively the real and imaginary parts of the complex number indicates. The absolute value of a complex number corresponds to the length of its vector in the Gaussian plane .
The amount norm is the standard scalar product of two real or complex numbers
- for or for
induced.
Vector norms
In the following, real or complex vectors of finite dimension are considered. A vector (in the narrower 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 of the following standards correspond to the amount standard of the previous section.
Maximum norm
The maximum norm, Chebyshev norm or ∞ norm (infinite norm) of a vector is defined as
and corresponds to the magnitude of the largest component of the vector. The unit sphere of the real maximum norm has the shape of a square in two dimensions, the shape of a cube in three dimensions and the shape of a hypercube in general dimensions .
The maximum norm is not induced by a scalar product. The metric derived from it is called the maximum metric , the Chebyshev metric or, especially in two dimensions, the checkerboard metric, as it measures the distance according to the number of steps a king in chess has to take to get from a square on the chessboard to come to another field. Since the king can move diagonally, for example the distance between the centers of the two diagonally opposite corner squares of a chessboard is the same in the maximum metric .
The maximum norm is a special case of the product norm
over the product space of normalized vector spaces with and .
Euclidean norm
The Euclidean norm or 2-norm of a vector is defined as
and corresponds to the root of the sum of the squares of the absolute values of the components of the vector. In the case of real vectors, the amount bars can be dispensed with in the definition, but not in the case of complex vectors.
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 general dimensions the shape of a sphere . The Euclidean norm describes the descriptive length of a vector in a plane or in space in two and three dimensions . The Euclidean norm is the only vector norm that is invariant under unitary transformations, for example rotations of the vector around the zero point.
The Euclidean norm is the standard scalar of two real or complex vectors given by
- or.
induced. A vector space provided with the Euclidean norm is called a Euclidean space . The metric derived from the Euclidean norm is called the Euclidean metric . For example, the distance between the center points of the two diagonally opposite corner fields of a chess board is the same in the Euclidean metric according to the Pythagorean theorem .
Sum norm
The sum norm, (more precisely) amount sum norm, or 1-norm (read: "one norm") of a vector is defined as
and corresponds to the sum of the amounts of the components of the vector. The unit sphere of the real sum norm has the shape of a square in two dimensions, the shape of an octahedron in three dimensions and the shape of a cross polytope in general dimensions .
The sum norm is not induced by a scalar product. The metric derived from the sum norm is also called Manhattan metric or taxi metric, especially in real two-dimensional space , since it measures the distance between two points like the route on a grid-like city map , on which one 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 is the same in the Manhattan metric .
p norms
In general, the -norm of a vector can be passed for real
define. For we obtain the sum norm for the Euclidean norm and set a limit for the maximum norm. The unit spheres of norms have in the real case in two dimensions in the form of Super ellipses or Subellipsen and in three and higher dimensions the form of Superellipsoiden or Subellipsoiden.
All norms including the maximum norm satisfy the Minkowski inequality as well as the Hölder inequality . They are monotonically decreasing for increasing and are equivalent to each other. The limiting factors for
- ,
where the exponent is set in the case of the maximum norm . The standards therefore differ by a maximum of the factor . The analogous to the norms for defined images are not norms, since the resulting norm spheres are no longer convex and the triangle inequality is violated.
Matrix norms
In the following, real or complex matrices with rows and columns are considered. For matrix norms, submultiplicativity is sometimes used in addition to the three norm properties
with required as a further defining property. If a matrix norm is sub-multiplicative, then the spectral radius of the matrix (the absolute value of the greatest eigenvalue) is at most as large as the norm of the matrix. However, there are also matrix norms with the usual norm properties that are not sub-multiplicative. In most cases, a vector norm is used as a basis when defining a matrix norm. A matrix norm is called compatible with a vector norm if
applies to all .
Matrix norms over vector norms
By writing all entries of a matrix one below the other, a matrix can also be viewed as a correspondingly long vector . This means that matrix norms can be defined directly using vector norms, in particular using the norms by
- ,
where the entries of the matrix are. Examples of matrix norms defined in this way are the overall norm based on the maximum norm and the Frobenius norm based on the Euclidean norm , both of which are sub-multiplicative and compatible with the Euclidean norm.
Matrix norms over operator norms
A matrix norm is called induced by a vector norm or a natural matrix norm if it is derived as an operator norm , so if:
- .
A matrix standard defined in this way clearly corresponds to the greatest possible expansion factor after applying the matrix to a vector. As operator norms, such matrix norms are always sub-multiplicative and compatible with the vector norm from which they were derived. An operator norm is even the one with the smallest value among all matrix norms compatible with a vector norm. Examples of matrix norms defined in this way are the row sum norm based on the maximum norm, the spectral norm based on the Euclidean norm and the column sum norm based on the sum norm .
Matrix norms over singular values
Another possibility to derive matrix norms via vector norms is to consider a singular value decomposition of a matrix into a unitary matrix , a diagonal matrix and an adjoint unitary matrix . The non-negative, real entries of are then the singular values of and equal to the square roots of the eigenvalues of . The singular values are then combined to form a vector whose vector norm is considered, i.e.
- .
Examples of matrix norms defined in this way are the shadow norms defined via the norms of the vector of singular values and the Ky-Fan norms based on the sum of the largest singular values .
Additional terms
Standardized spaces
If a vector space is provided with a norm, a normalized space with important analytical properties is obtained. Every norm between vectors induces a metric by forming the difference
- .
With this Fréchet metric , a normalized space becomes a metric space and, with the topology induced by the metric, it becomes a topological space , even a Hausdorff space . The standard is then a continuous mapping with respect to this standard topology . A sequence thus tends towards a limit value if and only if applies. If every Cauchy sequence converges in a normalized space to a limit value in this space, one speaks of a complete normalized space or Banach space .
Normalized algebras
If one also provides the vector space with an associative and distributive vector product , then it is an associative algebra . Is now a normed space and this standard submultiplicative, ie for all vectors applies
- ,
then one gets a normalized algebra. If the normalized space is complete, one speaks of a Banach algebra . For example, the space of the square matrices is the matrix addition and multiplication, and a matrix norm submultiplikativen such Banach.
Semi-norms
If the first norm axiom definiteness is dispensed with, then there is only a semi-norm (or a seminorm). Because of the homogeneity and the subadditivity, the amount is then
of the vectors with norm zero a subspace of . In this way, can equivalence relation on by
To be defined. If one now identifies all such equivalent elements as the same in a new space , then together with the norm there is a standardized space. This process is called residual class formation in relation to the semi-norm and referred to as a factor space . Special topological vector spaces , the locally convex spaces , can also be defined by a set of semi-norms .
Equivalence of norms
Two norms and are called equivalent if there are two positive constants and such that for all
applies, i.e. if one norm can be estimated upwards and downwards by the other norm. Equivalent norms induce the same topology. If a sequence converges with respect to a norm, then it also converges with respect to a norm equivalent to it.
On finite-dimensional vector spaces, all norms are equivalent to one another, since the norm spheres are then compact sets according to Heine-Borel's theorem . However, not all norms are equivalent to one another on infinite-dimensional spaces. Is a vector space but with respect to two standards completely , these two standards are equivalent if there is a positive constant are so
holds because there is a continuous linear mapping between the two Banach spaces, the inverse of which is also continuous according to the theorem of the continuous inverse .
Dual norms
The dual space of a normalized vector space over a body is the space of the continuous linear functionals from to . For example, the dual space to the space of the -dimensional (column) vectors can be seen as the space of the linear combinations of the vector components, that is to say the space of the row vectors of the same dimension. The dual norm of a functional is then defined by
- .
With this norm, the dual space is also a standardized space. The dual space with the dual norm is always complete, regardless of the completeness of the initial space. If two norms are equivalent to one another, then the associated dual norms are also equivalent to one another. For dual norms, the above definition as a supremum immediately results in the following important inequality
- .
Norms on infinite-dimensional vector spaces
Follow Norms
Real-valued or complex-valued sequences with sequence terms for are now considered. Consequences are thus a direct generalization of vectors of finite dimension. In contrast to finite-dimensional vectors, sequences can be unlimited, which means that the previous vector norms cannot be transferred directly to sequences. For example, the maximum amount or the total amount of the sequence members of an unlimited sequence is infinite and therefore no longer a real number. Therefore, the observed sequence spaces must be restricted accordingly so that the assigned norms are finite.
Supreme norm
The supremum norm of a restricted sequence is defined as
- .
The set of bounded sequences , the set of convergent sequences, and the set of sequences converging towards zero ( zero sequences ) are, together with the supreme norm, complete normalized spaces.
bv norm
The norm of a sequence of bounded variation is defined as
- .
With the -norm, the sequence space becomes a completely normalized space, since every sequence with bounded variation is a Cauchy sequence . For the subspace of zero sequences with limited variation, the -norm is obtained by omitting the first term, i.e.
- ,
and with this norm the space is also complete.
ℓ ^{p} norms
The norms are the generalization of the norms to sequence spaces, where only the finite sum is replaced by an infinite sum. The -norm of a sequence that can be summed in terms of magnitude is then defined for real as
- .
Provided with these norms, the spaces become complete standardized spaces. For the limit value there is the space of limited consequences with the supremum norm. The space is a Hilbert space with the scalar product
two episodes. The norm with a dual norm is the norm with . However, the space is not dual to space , but rather dual to the space of the convergent sequences and to the space of the zero sequences, each with the supremum norm.
Function standards
In the following, real or complex-valued functions on a set are considered. Often a topological space is used so that one can talk about continuity, in many applications a subset of the . Just like consequences, functions can in principle also be unlimited. Therefore, the function spaces considered must be restricted accordingly so that the assigned norms are finite. The most important such function spaces are classes of bounded, continuous, integrable or 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.
Supreme norm
The supremum norm of a bounded function, i.e. a function whose image is a bounded subset of , is defined as
- .
The set of restricted functions is a completely normalized space with the supreme norm.
BV standard
The -norm of a one-dimensional function with limited variation on an interval is defined by analogy with the -norm of a sequence as
- ,
where a partition of the interval and the supremum is taken over all possible partitions. A function is of limited variation if and only if it can be represented as the sum of a monotonically increasing and a monotonically decreasing function. The set of functions of limited variation is a complete normalized space with the -norm. Alternatively, instead of the integral of the function over the interval, the normalization term can be selected. There are a number of multidimensional generalizations for norms and the associated spaces of functions of limited variation, for example 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
- .
According to the extreme value theorem , a continuous function takes its maximum on a compact set. The space of continuous functions on a compact set is a completely normalized space with the maximum norm.
Holder norms
The Hölder norm of a Hölder continuous function with Höldere exponent is defined as
- ,
where the Hölder constant of the function by
given is. The Hölder constant is a special form of a continuity module and is itself a semi-norm. The spaces of the Hölder continuous functions are completely standardized spaces with the respective Hölder norms. In special cases one speaks of a Lipschitz continuous function, the Lipschitz constant and the Lipschitz norm.
Essential supremacy norm
The norm of an almost everywhere bounded function on a measure space is defined as
- ,
where is a null set , i.e. an element of the σ-algebra with a measure of zero. A function that is restricted almost everywhere can therefore at some points assume a value that is higher than its essential supremum. The essential supremum norm is generally only a semi-norm, since the set of functions with norm zero includes not only the null function , but also, for example, all functions that deviate from zero sets and assume values other than zero. Therefore we consider the set of equivalence classes of functions , which are almost everywhere the same, and call the corresponding factor space . In this space the essential supremacy norm is defined as
actually a norm, whereby the value on the right is independent of the choice of the representative from the equivalence class . Often it is written inaccurately instead of written, in which case it is assumed that only one representative of the equivalence class is. The space of the equivalence classes of essentially restricted functions is a completely standardized space with the essential supreme norm.
L ^{p} norms
The -norms of a Lebesgue-integrable function with a -th power are defined in analogy to the -norms as
- ,
where the sum has been replaced by an integral. As with the essential supreme norm, these norms are initially only semi-norms, since not only the null function, but also all functions that differ from the null function only in terms of a set with a measure of zero, are integrated to zero. Therefore we consider again the set of equivalence classes of functions that are almost the same everywhere, and defines this -spaces the norms by
- .
According to Fischer-Riesz's theorem , all -spaces with the respective -norm are complete normalized spaces. The space is the space of the (equivalence classes of) Lebesgue integrable functions. The space of the square integrable functions is a Hilbert space with a scalar product
and the space of the essentially restricted functions results for the limit value . The one to the norm for dual norm is the norm with . The -norms and -spaces can be generalized by the Lebesgue measure to general measures , whereby the duality only applies to certain measure spaces , see duality of L ^{p} -spaces . ^{}
C ^{m} standards
The -norm of a -time continuously differentiable function on an open set , whose partial derivatives can be continuously continued at the end of the set , is defined as
- ,
where is a multi-index of nonnegative integers, the associated mixed partial derivative of the function and the order of the derivative. The norm corresponds to the supreme norm and the norm to the maximum of the function and its first derivatives. The rooms are fully standardized rooms with the respective standard. Alternatively, the norm is defined using the sum of the individual norms instead of their maximum, but both norms are equivalent to one another.
Analogously, the -norm of a -time continuously differentiable function on an open set, whose mixed partial derivatives can be continuously continued at the end of the set and whose Hölder constants of the derivatives are limited to degree , is defined as
- .
The spaces of these Hölder continuously differentiable functions are also complete standardized spaces with the respective norms.
Sobolev norms
The Sobolev standard of times weak differentiable function on an open set, their mixed weak derivatives to degree in -th power are Lebesgue is integrated for a defined
and for as
- .
If you only consider the mixed derivatives of the order in the sum , you only get a semi-norm that disappears on all polynomials of degree less than zero. The Sobolev spaces of the functions, the mixed weak derivatives of which are up to the degree in , are completely normalized spaces with the respective Sobolev norm. In particular, the spaces are Hilbert spaces with a 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 in error estimates of finite element methods for the discretization of partial differential equations.
Norms on operators
In the following, linear operators between two vector spaces and are considered. It is assumed that these vector spaces are already normalized spaces themselves.
Operator norm
The operator norm of a bounded linear operator between two normalized spaces is defined as
- .
If there is a linear mapping between finitely dimensional vector spaces, then its operator norm is a natural matrix norm after choosing a basis. If the vector space is complete, then the space of the bounded (and thus continuous) linear operators from to is also complete. Operator norms are always sub-multiplicative, so if the two vector spaces are equal and complete, then the space of continuous linear operators with the operator norm and the composition is a Banach algebra.
Nuclear standard
The nuclear norm of a nuclear operator between two Banach spaces is defined as
- ,
where a sequence of vectors is in dual space and a sequence of vectors is in, so that has the shape , and the infimum is taken over all such nuclear representations. If the two vector spaces Hilbert spaces the corresponding nuclear standard will also track standard called. The space of the nuclear operators is a completely normalized space with the nuclear norm.
Hilbert-Schmidt standard
The Hilbert-Schmidt norm of a Hilbert-Schmidt operator between two Hilbert spaces is defined as
- ,
where is an orthonormal basis of . The Hilbert-Schmidt norm generalizes the Frobenius norm to the case of infinitely dimensional Hilbert spaces. The Hilbert-Schmidt norm is induced by the scalar product , where the adjoint operator is zu . The set of Hilbert-Schmidt operators with the Hilbert-Schmidt norm itself forms a Hilbert space and, for a Banach algebra, even an H * -algebra .
Shadow norms
The shadow norm of a compact linear operator between two separable Hilbert spaces is defined for as
- ,
where is the sequence of the singular values of the operator. In the case , the track norm results and in the case the Hilbert-Schmidt norm. The set of compact linear operators, whose singular values lie in, forms a complete normalized space with the respective shadow norm and for a Banach algebra.
Generalizations
Weighted norms
Weighted norms are norms on weighted vector spaces. For example, induced weighted function norms are obtained by multiplying by a suitable positive weight function over
- with ,
where is a weighted scalar product. The introduction of weighting functions makes it possible to expand function spaces, for example to functions whose norm would be unrestricted in the unweighted case, or to restrict them, for example to functions that exhibit a certain decay behavior.
Quasinorms
If the triangle inequality is weakened to the effect that there is only one real constant , so that for all
holds, the corresponding mapping is called a quasi- norm and a vector space provided with such a quasi-norm is called a quasi-normalized space. For example, the norms for quasinorms and the associated spaces are quasi-normalized spaces, even quasi-Banach spaces.
Scored bodies and modules
The concept of a norm can be understood more generally by allowing arbitrary vector spaces over evaluated bodies , i.e. bodies with an absolute value , instead of vector spaces over the body of real or complex numbers . Another generalization is that the vector space is replaced by a - (left) - module over a unitary ring of magnitude . A function is then called a norm on the module if the three norm properties definiteness, absolute homogeneity and subadditivity are fulfilled for all and all scalars . When the base ring of the amount by a pseudo amount is replaced in the module and the homogeneity is attenuated to Subhomogenität, one obtains a pseudo standard .
See also
literature
- Hans Wilhelm Alt: Linear Functional Analysis: An Application-Oriented Introduction . 7th edition. Springer-Verlag, 2007, ISBN 3-540-34187-0 .
- Albrecht Beutelspacher : Linear Algebra. An introduction to the science of vectors, maps, and matrices . 5th edition. Vieweg, 2009, ISBN 3-528-66508-4 .
- Michel M. Deza, Elena Deza: Encyclopedia of Distances . Springer, 2009, ISBN 978-3-642-00233-5 .
- Gerd Fischer : Linear Algebra: An Introduction for New Students . 17th edition. Vieweg Verlag, 2009, ISBN 3-8348-0996-9 .
- Hans Rudolf Schwarz, Norbert Köckler: Numerical Mathematics . 8th edition. Vieweg & Teubner, 2011, ISBN 978-3-8348-1551-4 .
- Dirk Werner : Functional Analysis . Springer-Verlag, 2007, ISBN 978-3-540-72533-6 .
Web links
- Eric W. Weisstein : Norm . In: MathWorld (English).
- EA Gorin: Norm . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
Individual evidence
- ↑ Harro Heuser : Textbook of Analysis . Part 2. 13th edition. Teubner Verlag, 2004, ISBN 3-519-62232-7 , pp. 19th f .
- ^ Stefan Banach: Sur les opérations dans les ensembles abstraits et leur application aux equations intégrales . In: Fundamenta Mathematicae . No. 3 , 1922.
- ^ Werner: Functional Analysis . Springer, 2007, p. 41 .
- ↑ ^{a } ^{b} Scriba, Schreiber: 5000 years of geometry: history, cultures, people . Springer, 2009, p. 511-512 .
- ^ Hans Wilhelm Alt: Linear functional analysis . 5th edition. Springer, 2007, p. 11 .
- ↑ ^{a } ^{b } ^{c } ^{d} Hans Wilhelm Alt: Lineare functional analysis . 5th edition. Springer, 2007, p. 26-27 .
- ^ Hans Wilhelm Alt: Linear functional analysis . 5th edition. Springer, 2007, p. 12 .
- ^ Hans Wilhelm Alt: Linear functional analysis . 5th edition. Springer, 2007, p. 20 .
- ^ Hans Wilhelm Alt: Linear functional analysis . 5th edition. Springer, 2007, p. 132 .
- ^ Hans Wilhelm Alt: Linear functional analysis . 5th edition. Springer, 2007, p. 36 .
- ^ Hans Wilhelm Alt: Linear functional analysis . 5th edition. Springer, 2007, p. 190 .
- ^ Hans Wilhelm Alt: Linear functional analysis . 5th edition. Springer, 2007, p. 39 .
- ^ Hans Wilhelm Alt: Linear functional analysis . 5th edition. Springer, 2007, p. 43-44 .
- ^ Hans Wilhelm Alt: Linear functional analysis . 5th edition. Springer, 2007, p. 49 .
- ^ Hans Wilhelm Alt: Linear functional analysis . 5th edition. Springer, 2007, p. 49-50 .
- ^ Hans Wilhelm Alt: Linear functional analysis . 5th edition. Springer, 2007, p. 41-43 .
- ^ Hans Triebel: Higher Analysis , Deutscher Verlag der Wissenschaften, Berlin 1972, 2nd edition, Harri Deutsch 1980, ISBN 3-87144-583-5 , comment 3.4
- ^ Hans Wilhelm Alt: Linear functional analysis . 5th edition. Springer, 2007, p. 62-65 .
- ↑ Michel M. Deza, Elena Deza: Encyclopedia of Distances . Springer, 2009, p. 236 .
- ↑ Michel M. Deza, Elena Deza: Encyclopedia of Distances . Springer, 2009, p. 236-237 .
- ↑ Michel M. Deza, Elena Deza: Encyclopedia of Distances . Springer, 2009, p. 237-238 .
- ↑ Michel M. Deza, Elena Deza: Encyclopedia of Distances . Springer, 2009, p. 238 .
- ↑ Falko Lorenz: Introduction to Algebra II . 2nd Edition. Spectrum Academic Publishing House, 1997, p. 69 .