Inner product space - Citizendia

For the scalar product or dot product of spatial vectors, see dot product. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R

Geometric interpretation of inner product

In mathematics, an inner product space is a vector space of arbitrary (possibly infinite) dimension with additional structure, which, among other things, enables generalization of concepts from two or three-dimensional Euclidean geometry. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Structure is a fundamental and sometimes Intangible notion covering the Recognition, Observation, nature, and Stability of Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. The additional structure associates to each pair of vectors in the space a number which is called the inner product (also called a scalar product) of the vectors. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R Inner products allow the rigorous introduction of intuitive geometrical notions such as the angle between vectors or length of vectors in spaces of all dimensions. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called Length is the long Dimension of any object The length of a thing is the distance between its ends its linear extent as measured from end to end It also allows introduction of the concept of orthogonality between vectors. In Mathematics, two Vectors are orthogonal if they are Perpendicular, i Inner product spaces generalize Euclidean spaces (with the dot product as the inner product) and are studied in functional analysis. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R For functional analysis as used in psychology see the Functional analysis (psychology article

An inner product space is sometimes also called a pre-Hilbert space, since its completion with respect to the metric, induced by its inner product, is a Hilbert space. In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined 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 This article assumes some familiarity with Analytic geometry and the concept of a limit.

Inner product spaces were referred to as unitary spaces in earlier work, although this terminology is now rarely used.

1 Definition
2 Examples
3 Norms on inner product spaces
4 Orthonormal sequences
5 Operators on inner product spaces
6 Degenerate inner products
7 See also
8 References

Definition

In the following article, the field of scalars denoted F is either the field of real numbers R or the field of complex numbers C. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted See below.

Formally, an inner product space is a vector space V over the field F together with a positive-definite sesquilinear form, called an inner product. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, a definite bilinear form is a Bilinear form B such that B ( x, x) has a fixed In Mathematics, a sesquilinear form on a Complex vector space V is a map V × V &rarr C that is linear In Multilinear algebra, a multilinear form is a map of the type f V^N \to K where V is a Vector space For real vector spaces, this is actually a positive-definite symmetric bilinear form. A symmetric bilinear form is as the name implies a Bilinear form on a Vector space that is symmetric In Mathematics, a bilinear form on a Vector space V is a Bilinear mapping V × V → F, where Thus the inner product is a map

$\langle \cdot, \cdot \rangle : V \times V \rightarrow \mathbb{F}$

satisfying the following axioms for all $x,y,z \in V, a,b \in \mathbb{F}$ :

Conjugate symmetry:

$\langle x,y\rangle =\overline{\langle y,x\rangle}.$

This condition implies that $\langle x,x\rangle \in \mathbb{R}$ , because $\langle x,x\rangle = \overline{\langle x,x\rangle}$ . In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject In Mathematics, the complex conjugate of a Complex number is given by changing the sign of the Imaginary part.

(Conjugation is also often written with an asterisk, as in $\langle y,x\rangle^{*}$ , as is the conjugate transpose. In Mathematics, the conjugate transpose, Hermitian transpose, or adjoint matrix of an m -by- n matrix A with )

Linearity in the first variable:

$\langle ax,y\rangle= a \langle x,y\rangle.$

$\langle x+y,z\rangle= \langle x,z\rangle+ \langle y,z\rangle.$

By combining these with conjugate symmetry, we get:

$\langle x,by \rangle= \overline{b} \langle x,y\rangle.$

$\langle x,y+z\rangle= \langle x,y\rangle+ \langle x,z\rangle.$

So $\langle \cdot , \cdot \rangle$ is actually a sesquilinear form. The word linear comes from the Latin word linearis, which means created by lines. In Mathematics, a sesquilinear form on a Complex vector space V is a map V × V &rarr C that is linear

Positivity:

$\langle x,x\rangle > 0$ for all $x \ne 0$ .

(This makes sense because $\langle x,x\rangle \in \mathbb{R}$ for all $x\in V$ . )

Definiteness:

$\langle x,x \rangle = 0$ implies

x = 0

Hence, the inner product is a positive-definite Hermitian form. In Mathematics, a sesquilinear form on a Complex vector space V is a map V × V &rarr C that is linear

The property of an inner product space

V

that

$\langle x+y,z\rangle= \langle x,z\rangle+ \langle y,z\rangle$ and $\langle x,y+z\rangle = \langle x,y\rangle + \langle x,z\rangle$ is known as additivity.

Note that if F=R, then the conjugate symmetry property is simply symmetry of the inner product, i. e. ,

$\langle x,y\rangle=\langle y,x\rangle.$

In this case, sesquilinearity becomes standard bilinearity. In Mathematics, a bilinear map is a function of two arguments that is linear in each

Remark: In the more abstract linear algebra literature, the conjugate-linear argument of the inner product is conventionally put in the second position (e. g. , y in $\langle x,y\rangle$ ) as we have done above. This convention is reversed in both physics and matrix algebra; in those respective disciplines we would write the product $\langle x,y\rangle$ as $\langle y|x\rangle$ (the bra-ket notation of quantum mechanics) and $y T x$ (dot product as a case of the convention of forming the matrix product AB as the dot products of rows of A with columns of B). Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Bra-ket notation is a standard notation for describing Quantum states in the theory of Quantum mechanics composed of angle brackets (chevrons and Vertical Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons Here the kets and columns are identified with the vectors of V and the bras and rows with the dual vectors or linear functionals of the dual space V^*, with conjugacy associated with duality. This article deals with Linear maps from a Vector space to its field of scalars These maps may be functionals in the traditional This article deals with Linear maps from a Vector space to its field of scalars These maps may be functionals in the traditional In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals This reverse order is now occasionally followed in the more abstract literature, e. g. , Emch [1972], taking $\langle x,y\rangle$ to be conjugate linear in x rather than y. A few instead find a middle ground by recognizing both < , > and < | > as distinct notations differing only in which argument is conjugate linear.

There are various technical reasons why it is necessary to restrict the basefield to R and C in the definition. In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication Briefly, the basefield has to contain an ordered subfield (in order for non-negativity to make sense) and therefore has to have characteristic equal to 0. In Mathematics, an ordered field is a field together with a Total ordering of its elements that agrees in a certain sense with the field operations In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's This immediately excludes finite fields. The basefield has to have additional structure, such as a distinguished automorphism. More generally any quadratically closed subfield of R or C will suffice for this purpose, e. g. , the algebraic numbers, but when it is a proper subfield (i. In Mathematics, an algebraic number is a Complex number that is a root of a non-zero Polynomial in one variable with rational (or e. , neither R nor C) even finite-dimensional inner product spaces will fail to be metrically complete. In contrast all finite-dimensional inner product spaces over R or C, such as those used in quantum computation, are automatically metrically complete and hence Hilbert spaces. A quantum computer is a device for Computation that makes direct use of distinctively Quantum mechanical Phenomena, such as superposition

In some cases we need to consider non-negative semi-definite sesquilinear forms. This means that <x, x> is only required to be non-negative. We show how to treat these below.

Examples

A trivial example is the real numbers with the standard multiplication as the inner product

$\langle x,y\rangle := xy$

More generally any Euclidean space Rⁿ with the dot product is an inner product space

$\langle (x_1,\ldots, x_n),(y_1,\ldots, y_n)\rangle := \sum_{i=1}^{n} x_i y_i = x_1 y_1 + \cdots + x_n y_n$

The general form of an inner product on Cⁿ is given by:

$\langle \mathbf{x},\mathbf{y}\rangle := \mathbf{x}^*\mathbf{M}\mathbf{y}$

with M any symmetric positive-definite matrix, and x^* the conjugate transpose of x. In Mathematics, the real numbers may be described informally in several different ways In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R In Linear algebra, a positive-definite matrix is a (Hermitian matrix which in many ways is analogous to a Positive Real number. In Mathematics, the conjugate transpose, Hermitian transpose, or adjoint matrix of an m -by- n matrix A with For the real case this corresponds to the dot product of the results of directionally differential scaling of the two vectors, with positive scale factors and orthogonal directions of scaling. In Euclidean geometry, uniform scaling or Isotropic scaling is a Linear transformation that enlarges or diminishes objects the Scale factor A scale factor is a number which scales, or multiplies some quantity Apart from an orthogonal transformation it is a weighted-sum version of the dot product, with positive weights. A weight function is a mathematical device used when performing a sum integral or average in order to give some elements more of a "weight" than others

The article on Hilbert space has several examples of inner product spaces wherein the metric induced by the inner product yields a complete metric space. This article assumes some familiarity with Analytic geometry and the concept of a limit. In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has An example of an inner product which induces an incomplete metric occurs with the space C[a, b] of continuous complex valued functions on the interval [a, b]. The inner product is

$\langle f , g \rangle := \int_a^b f(t) \overline{g(t)} \, dt$

This space is not complete; consider for example, for the interval [−1,1] the sequence of "step" functions { f_k }_k where

f_k(t) is 0 for t in the subinterval [−1,0]
f_k(t) is 1 for t in the subinterval [1/k, 1]
f_k is affine in [0, 1/k]

This sequence is a Cauchy sequence which does not converge to a continuous function.

Norms on inner product spaces

Inner product spaces have a naturally defined norm

$\|x\| =\sqrt{\langle x, x\rangle}.$

This is well defined by the nonnegativity axiom of the definition of inner product space. 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 The norm is thought of as the length of the vector x. Directly from the axioms, we can prove the following:

Cauchy-Schwarz inequality: for x, y elements of V

$|\langle x,y\rangle| \leq \|x\| \cdot \|y\|$

with equality if and only if x and y are linearly dependent. In Mathematics, the Cauchy–Schwarz inequality, also known as the Schwarz inequality, the Cauchy inequality, or the Cauchy–Schwarz–Bunyakovsky In Linear algebra, a family of vectors is linearly independent if none of them can be written as a Linear combination of finitely many other vectors This is one of the most important inequalities in mathematics. It is also known in the Russian mathematical literature as the Cauchy-Bunyakowski-Schwarz inequality.

Because of its importance, its short proof should be noted. To prove this inequality note it is trivial in the case y = 0. Thus we may assume <y, y> is nonzero. Thus we may let

$\lambda = \langle y , y \rangle^{-1} \langle x, y\rangle$

and it follows that

$0 \leq \langle x -\lambda y, x -\lambda y \rangle = \langle x, x\rangle - \langle y , y \rangle^{-1} | \langle x,y\rangle|^2.$

multiplying out, the result follows.

Orthogonality: The geometric interpretation of the inner product in terms of angle and length, motivates much of the geometric terminology we use in regard to these spaces. In Mathematics, two Vectors are orthogonal if they are Perpendicular, i Indeed, an immediate consequence of the Cauchy-Schwarz inequality is that it justifies defining the angle between two non-zero vectors x and y (at least in the case F = R) by the identity

$\operatorname{angle}(x,y) = \arccos \frac{\langle x, y \rangle}{\|x\| \cdot \|y\|}.$

We assume the value of the angle is chosen to be in the interval [0, +π]. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called This is in analogy to the situation in two-dimensional Euclidean space. Correspondingly, we will say that non-zero vectors x, y of V are orthogonal if and only if their inner product is zero.

Homogeneity: for x an element of V and r a scalar

$\|r \cdot x\| = |r| \cdot \| x\|.$

The homogeneity property is completely trivial to prove. In Mathematics, a homogeneous function is a function with multiplicative scaling behaviour if the argument is multiplied by a factor then the result is multiplied by some power

Triangle inequality: for x, y elements of V

$\|x + y\| \leq \|x \| + \|y\|.$

The last two properties show the function defined is indeed a norm. In Mathematics, the triangle inequality states that for any Triangle, the length of a given side must be less than or equal to the sum of the other two sides but greater

Because of the triangle inequality and because of axiom 2, we see that ||·|| is a norm which turns V into a normed vector space and hence also into a metric space. 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 In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined The most important inner product spaces are the ones which are complete with respect to this metric; they are called Hilbert spaces. In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has This article assumes some familiarity with Analytic geometry and the concept of a limit. Every inner product V space is a dense subspace of some Hilbert space. In Topology and related areas of Mathematics, a Subset A of a Topological space X is called dense (in X) if This Hilbert space is essentially uniquely determined by V and is constructed by completing V.

Parallelogram law: for x, y elements of V,

$\|x + y\|^2 + \|x - y\|^2 = 2\|x\|^2 + 2\|y\|^2.$

Pythagorean theorem: Whenever x, y are in V and <x, y> = 0, then

$\|x\|^2 + \|y\|^2 = \|x+y\|^2.$

The proofs of both of these identities require only expressing the definition of norm in terms of the inner product and multiplying out, using the property of additivity of each component. In Mathematics, the simplest form of the parallelogram law belongs to elementary Geometry. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry Alternatively, both can be seen as consequences of the identity

$\|x + y\|^2 = \|x\|^2 + \|y\|^2 + 2Re \langle x , y \rangle.$

which is a form of the law of cosines, and is proved as before. In Trigonometry, the law of cosines (also known as Al-Kashi law or the cosine formula or cosine rule) is a statement about a general

The name Pythagorean theorem arises from the geometric interpretation of this result as an analogue of the theorem in synthetic geometry. Synthetic geometry is the branch of Geometry which makes use of Theorems and synthetic observations to draw conclusions as opposed to Analytic geometry Note that the proof of the Pythagorean theorem in synthetic geometry is considerably more elaborate because of the paucity of underlying structure. In this sense, the synthetic Pythagorean theorem, if correctly demonstrated is deeper than the version given above.

An easy induction on the Pythagorean theorem yields:

If x₁, . Mathematical induction is a method of Mathematical proof typically used to establish that a given statement is true of all Natural numbers It is done by proving that . . , x_n are orthogonal vectors, that is, <x_j, x_k> = 0 for distinct indices j, k, then

$\sum_{i=1}^n \|x_i\|^2 = \left\|\sum_{i=1}^n x_i \right\|^2.$

In view of the Cauchy-Schwarz inequality, we also note that <·,·> is continuous from V × V to F. In Mathematics, two Vectors are orthogonal if they are Perpendicular, i In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output This allows us to extend Pythagoras' theorem to infinitely many summands:

Parseval's identity: Suppose V is a complete inner product space. If {x_k} are mutually orthogonal vectors in V then

$\sum_{i=1}^\infty\|x_i\|^2 = \left\|\sum_{i=1}^\infty x_i\right\|^2,$

provided the infinite series on the left is convergent. In the absence of a more specific context convergence denotes the approach toward a definite value as time goes on or to a definite point a common view or opinion or Completeness of the space is needed to ensure that the sequence of partial sums

$S_k = \sum_{i=1}^k x_i$

which is easily shown to be a Cauchy sequence is convergent. In Mathematics, a Cauchy sequence, named after Augustin Cauchy, is a Sequence whose elements become arbitrarily close to each other as the sequence

Orthonormal sequences

A sequence {e_k}_k is orthonormal if and only if it is orthogonal and each e_k has norm 1. ↔ An orthonormal basis for an inner product space of finite dimension V is an orthonormal sequence whose algebraic span is V. This definition of orthonormal basis does not generalise conveniently to the case of infinite dimensions, where the concept (properly formulated) is of major importance. Using the norm associated to the inner product, one has the notion of dense subset, and the appropriate definition of orthonormal basis is that the algebraic span (subspace of finite linear combinations of basis vectors) should be dense. In Topology and related areas of Mathematics, a Subset A of a Topological space X is called dense (in X) if

The Gram-Schmidt process is a canonical procedure that takes a linearly independent sequence {v_k}_k on an inner product space and produces an orthonormal sequence {e_k}_k such that for each n

$\operatorname{span}\{v_1, \ldots, v_n\} = \operatorname{span}\{e_1, \ldots, e_n\}$

By the Gram-Schmidt orthonormalization process, one shows:

Theorem. In Mathematics, particularly Linear algebra and Numerical analysis, the Gram–Schmidt process is a method for orthogonalizing a set of Any separable inner product space V has an orthonormal basis. In Mathematics a Topological space is called separable if it contains a countable dense subset that is there exists a sequence \{ x_n

Parseval's identity leads immediately to the following theorem:

Theorem. Let V be a separable inner product space and {e_k}_k an orthonormal basis of V. Then the map

$x \mapsto \{\langle e_k, x\rangle\}_{k \in \mathbb{N}}$

is an isometric linear map V → l² with a dense image.

This theorem can be regarded as an abstract form of Fourier series, in which an arbitrary orthonormal basis plays the role of the sequence of trigonometric polynomials. In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions In the Mathematical subfields of Numerical analysis and Mathematical analysis, a trigonometric polynomial is a finite Linear combination of sin( Note that the underlying index set can be taken to be any countable set (and in fact any set whatsoever, provided l² is defined appropriately, as is explained in the article Hilbert space). This article assumes some familiarity with Analytic geometry and the concept of a limit. In particular, we obtain the following result in the theory of Fourier series:

Theorem. Let V be the inner product space $C [ - π,π]$ . Then the sequence (indexed on set of all integers) of continuous functions

e k (t) = (2π) - 1 / 2 e i k t

is an orthonormal basis of the space $C [ - π,π]$ with the L² inner product. The mapping

$f \mapsto \frac{1}{\sqrt{2 \pi}} \left\{\int_{-\pi}^\pi f(t) e^{-i k t} \, dt \right\}_{k \in \mathbb{Z}}$

is an isometric linear map with dense image.

Orthogonality of the sequence {e_k}_k follows immediately from the fact that if k ≠ j, then

$\int_{-\pi}^\pi e^{-i (j-k) t} \, dt = 0.$

Normality of the sequence is by design, that is, the coefficients are so chosen so that the norm comes out to 1. Finally the fact that the sequence has a dense algebraic span, in the inner product norm, follows from the fact that the sequence has a dense algebraic span, this time in the space of continuous periodic functions on $[ - π,π]$ with the uniform norm. This is the content of the Weierstrass theorem on the uniform density of trigonometric polynomials.

Operators on inner product spaces

Several types of linear maps A from an inner product space V to an inner product space W are of relevance:

Continuous linear maps, i. The word linear comes from the Latin word linearis, which means created by lines. In Functional analysis and related areas of Mathematics, a continuous linear operator or continuous linear mapping is a continuous Linear e. A is linear and continuous with respect to the metric defined above, or equivalently, A is linear and the set of non-negative reals {||Ax||}, where x ranges over the closed unit ball of V, is bounded.
Symmetric linear operators, i. e. A is linear and <Ax, y> = <x, A y> for all x, y in V.
Isometries, i. e. A is linear and <Ax, Ay> = <x, y> for all x, y in V, or equivalently, A is linear and ||Ax|| = ||x|| for all x in V. All isometries are injective. Isometries are morphisms between inner product spaces, and morphisms of real inner product spaces are orthogonal transformations (compare with orthogonal matrix). In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In Matrix theory, a real orthogonal matrix is a square matrix Q whose Transpose is its inverse: Q^T
Isometrical isomorphisms, i. e. A is an isometry which is surjective (and hence bijective). In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property Isometrical isomorphisms are also known as unitary operators (compare with unitary matrix). In Mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition U^* U = UU^*

From the point of view of inner product space theory, there is no need to distinguish between two spaces which are isometrically isomorphic. The spectral theorem provides a canonical form for symmetric, unitary and more generally normal operators on finite dimensional inner product spaces. In Mathematics, particularly Linear algebra and Functional analysis, the spectral theorem is any of a number of results about Linear operators In Mathematics, especially Functional analysis, a normal operator on a Hilbert space H (or more generally in a C* algebra) is a A generalization of the spectral theorem holds for continuous normal operators in Hilbert spaces.

Degenerate inner products

If V is a vector space and < , > a semi-definite sesquilinear form, then the function ||x|| = <x, x>^1/2 makes sense and satisfies all the properties of norm except that ||x|| = 0 does not imply x = 0. (Such a functional is then called a semi-norm. In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length ) We can produce an inner product space by considering the quotient W = V/{ x : ||x|| = 0}. The sesquilinear form < , > factors through W.

This construction is used in numerous contexts. The Gelfand-Naimark-Segal construction is a particularly important example of the use of this technique. In Functional analysis, given a C*-algebra A, the Gelfand-Naimark-Segal construction establishes a correspondence between cyclic *-representations of Another example is the representation of semi-definite kernels on arbitrary sets.

References

Axler, Sheldon (1997), Linear Algebra Done Right (2nd ed. In Linear algebra, the outer product typically refers to the tensor product of two vectors. In Mathematics, a bilinear form on a Vector space V is a Bilinear mapping V × V → F, where In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals In Functional analysis and related areas of Mathematics a dual pair or dual system is a pair of Vector spaces with an associated Bilinear In Mathematics, a biorthogonal system is a pair of Topological vector spaces E and F that are in duality, with a pair of indexed subsets In Mathematics, the Fubini-Study metric is a Kähler metric on Projective Hilbert space, that is Complex projective space CP In Mathematics, more precisely in Functional analysis, an energetic space is intuitively a subspace of a given real Hilbert space equipped ), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98258-8
Emch, Gerard G. Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic (1972), Algebraic methods in statistical mechanics and quantum field theory, Wiley-Interscience, ISBN 978-0-471-23900-0
Young, Nicholas (1988), An introduction to Hilbert space, Cambridge University Press, ISBN 978-0-521-33717-5

John Wiley & Sons Inc, also referred to as Wiley, is a global Publishing company that markets its products to professionals and consumers students and instructors Cambridge University Press (known colloquially as CUP is a Publisher given a Royal Charter by Henry VIII in 1534

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