The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. In more formal terms, a Hilbert space is an inner product space — an abstract vector space in which distances and angles can be measured — which is "complete", meaning that if a sequence of vectors approaches a limit, then that limit is guaranteed to be in the space as well. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... David Hilbert (January 23, 1862, Königsberg, East Prussia – February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... Partial plot of a function f. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ...

Hilbert spaces arise naturally and frequently in mathematics, physics, and engineering, typically as infinite-dimensional function spaces. They are indispensable tools in the theories of partial differential equations, quantum mechanics, and signal processing. The recognition of a common algebraic structure within these diverse fields generated a greater conceptual understanding, and the success of Hilbert space methods ushered in a very fruitful era for functional analysis. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the fundamental laws of the universe. ... Engineering is the design, analysis, and/or construction of works for practical purposes. ... In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications, it is a topological space or a vector space or both. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics. ... Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...

Geometric intuition plays an important role in many aspects of Hilbert space theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to an orthonormal basis, in analogy with cartesian coordinates in the plane. This means that Hilbert space can also usefully be thought of in terms of infinite sequences that are square summable. Linear operators on a Hilbert space are likewise fairly concrete objects: in good cases, they are simply transformations that stretch the space by different factors in mutually perpendicular directions. In mathematics, an orthonormal basis of an inner product space V(i. ... This is a page about mathematics. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...

Motivation and intuitive meaning

Ordinary Euclidean space R³ serves as a model for the more abstract notion of a Hilbert space. In the Euclidean space, the distance between points and the angle between vectors can be expressed via the dot product, a certain bilinear operation on vectors with values in real numbers. Many problems from analytic geometry can be reworded and solved using the dot product, for example, "When are two lines orthogonal?" or "How to find the point on a given plane closest to the origin?" Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... Distance is a numerical description of how far apart objects are at any given moment in time. ... An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ... In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ... Please refer to Real vs. ... Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra. ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...

In a Hilbert space, the fundamental objects are abstractions of vectors, whose nature is unimportant (they may be, for example, sequences or functions of some kind). Those abstract vectors can be added and multiplied by a scalar, and an analogue of the dot product is defined for them. The algebraic operations on vectors in a Hilbert space have familiar properties, like commutativity and distributivity. In addition, the technical requirement of completeness ensures that certain limits exist. This last property is always true for finite-dimensional inner product spaces, but needs to be stated as an additional assumption in the more general case. Completeness guarantees that various geometric operations, such as orthogonal projection onto a subspace, that are familiar in the setting of Euclidean spaces, can be meaningfully defined in general, even for an infinite dimensional space. A map or binary operation from a set to a set is said to be commutative if, (A common example in school-math is the + function: , thus the + function is commutative) Otherwise, the operation is noncommutative. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In geometry, an orthogonal projection of a k-dimensional object onto a d-dimensional hyperplane (d < k) is obtained by intersections of (k &#8722; d)- dimensional hyperplanes drawn through the points of an object orthogonally to the d-hyperplane. ...

While the definition of a Hilbert space given below may appear complicated, due to a large number of consistency axioms, the basic intuition behind Hilbert spaces is amazingly simple:

In a large range of physical and mathematical situations, a linear problem can be stated within a certain Hilbert space and analyzed in simple geometrical terms.

In particular, this principle applies to solving linear differential and integral equations, and especially eigenvalue problems. One of the first examples of such an analysis was given by Joseph Fourier's mathematical theory of heat: a solution of the heat equation can be decomposed into infinitely many independent parts, which is closely analogous to the way of representing a vector from R³ as a linear combination of three orthogonal vectors. Similar considerations apply to other equations of mathematical physics, notably, the wave equation and Helmholtz equation. In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... Jean Baptiste Joseph Fourier (March 21, 1768 - May 16, 1830) was a French mathematician and physicist who is best known for initiating the investigation of Fourier series and their application to problems of heat flow. ... The heat equation is an important partial differential equation which describes the variation of temperature in a given region over time. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... The wave equation is an important partial differential equation that describes a variety of waves, such as sound waves, light waves and water waves. ... The Helmholtz equation, named for Hermann von Helmholtz, is the following elliptic partial differential equation: The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. ...

The success of the theory of Hilbert spaces is due in part to the striking fact that

although they may differ in origin and appearance, most Hilbert spaces considered in physics and mathematics are just multiple manifestations of a single separable Hilbert space.

One way to comprehend this proceeds by introducing a system of coordinates into a given Hilbert space using the notion of orthonormal basis described below. As a consequence of the uniqueness principle, a theorem stated in abstract terms and valid in one of these spaces will hold in all of them. In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space. ... In mathematics, an orthonormal basis of an inner product space V(i. ...

Definition

A real or complex Hilbert space is a real or complex inner product space that is a complete normed space (Banach space) under the norm defined by the inner product. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...

Remarks

1. The inner product 〈·,·〉 on a real or complex vector space H gives rise to a norm ||·|| as follows:

   

2. Completeness is the key to handling infinite-dimensional examples, such as function spaces. It is expressed using a form of the Cauchy criterion for sequences in H:

   A sequence {v_n} is a Cauchy sequence if for every positive real number ε there is a natural number N such that for all m, n > N, ||v_n – v_m|| < ε. The space H is complete with respect to this norm if every Cauchy sequence converges to an element in the space.

3. As any normed vector space, an inner product space becomes a topological vector space if we declare that the open balls constitute a basis of topology. A Hilbert space is also a Banach space in which the following parallelogram identity holds:

   

4. Conversely, it can be proved that a Banach space in which the parallelogram identity holds is a Hilbert space, and the inner product is uniquely determined by the norm.
5. Some authors use slightly different definitions. For example, Kolmogorov-Fomin^[1] define a Hilbert space as above but restrict the definition to separable and infinite-dimensional spaces. A separable, infinite-dimensional Hilbert space is unique up to isomorphism, called [often written for shorthand — see the next section for the definition]. In this article, a Hilbert space is not assumed to be infinite-dimensional or separable.
6. Older books and papers sometimes call a Hilbert space a unitary space or a linear space with an inner product, but this terminology fell out of use.

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, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ... The Cauchy convergence test is a method used to test infinite series for convergence. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ... In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ... A synonym for ball (in geometry or topology, and in any dimension) is disk (or disc Geometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... Separable can refer to: Separable space in topology Separable sigma algebra in measure theory Separable differential equations This is a disambiguation page &#8212; a navigational aid which lists other pages that might otherwise share the same title. ...

Genesis of Hilbert spaces

The first important theorems that apply to Hilbert spaces were obtained by Joseph Fourier, Friedrich Bessel and Marc-Antoine Parseval in the 19th century in the context of periodic functions of one real variable. Fourier's theory of trigonometric series in particular provides a template for the later development of the theory of function spaces in an abstract setting. Further basic results were proved in early 20th century, for example, the Riesz representation theorem of Maurice Frechet and Frigyes Riesz from 1907. Jean Baptiste Joseph Fourier (March 21, 1768 - May 16, 1830) was a French mathematician and physicist who is best known for initiating the investigation of Fourier series and their application to problems of heat flow. ... Friedrich Wilhelm Bessel (July 22, 1784 – March 17, 1846) was a German mathematician, astronomer, and systematizer of the Bessel functions (which, despite their name, were discovered by Daniel Bernoulli). ... Marc-Antoine Parseval des Chênes (April 27, 1755 &#8211; August 16, 1836) was a French mathematician, most famous for what is now known as Parsevals theorem, which presaged the unitarity of the Fourier transform. ... In mathematics, a Fourier series of a periodic function, named in honor of Joseph Fourier (1768-1830), represents the function as a sum of periodic functions of the form where e is Eulers number and i the imaginary unit. ... There are several well-known theorems in functional analysis known as the Riesz representation theorem. ... Maurice Fréchet (born September 2, 1878, died June 4, 1973) was a French mathematician. ... Frigyes Riesz Frigyes Riesz (January 22, 1880 – February 28, 1956) was a mathematician who was born in GyÅ‘r, Austria-Hungary (now Hungary) and died in Budapest Hungary. ...

Hilbert spaces are named after David Hilbert, who developed methods of infinite-dimensional linear algebra in the course of his work on integral equations beginning around 1909. Hilbert's axiomatic approach to the study of function spaces and operators on them, which may be termed the "algebraization of analysis", provided the foundations for functional analysis as a new mathematical discipline, and made profound impact on the later development of mathematics. David Hilbert (January 23, 1862, Königsberg, East Prussia – February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ... In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...

The significance of the concept of Hilbert space was underlined with the realization that it offers one of the best mathematical formulations of quantum mechanics. In short, the states of a quantum mechanical system are described by vectors in a certain Hilbert space, the observables are expressed by linear operators, and the procedure of quantum measurement is related to orthogonal projection. Moreover, the symmetries of a quantum mechanical system can be interpreted as a unitary representation of a suitable group, providing an impetus for development of unitary representation theory. On the other hand, around the same time it became clear that certain properties of classical dynamical systems can be analyzed using Hilbert space techniques in the framework of ergodic theory.^{[citation needed]} The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications. ... In mathematics, a unitary representation of a group G is a linear representation &#960; of G on a complex Hilbert space V such that &#960;(g) is a unitary operator for every g &#8712; G. The general theory is well-developed in case G is a locally compact (Hausdorff) topological... This picture illustrates how the hours in a clock form a group. ... In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ... In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ... In mathematics, a measure-preserving transformation T on a probability space is said to be ergodic if the only measurable sets invariant under T have measure 0 or 1. ...

John von Neumann coined the term abstract Hilbert space in his famous work on unbounded Hermitian operators, published in 1929.^[2] Von Neumann was perhaps the mathematician who most clearly recognized their importance as a result of his seminal work on the foundations of quantum mechanics which began with Hilbert and Lothar (Wolfgang) Nordheim^[3] and continued with Eugene Wigner. The name "Hilbert space" was soon adopted by others, for example by Hermann Weyl in his 1931 book The Theory of Groups and Quantum Mechanics.^[4] John von Neumann (Hungarian Margittai Neumann János Lajos) (born December 28, 1903 in Budapest, Austria-Hungary; died February 8, 1957 in Washington D.C., United States) was a Hungarian-born mathematician and polymath who made contributions to quantum physics, functional analysis, set theory, topology, economics, computer science, numerical analysis... On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ... 1929 (MCMXXIX) was a common year starting on Tuesday (link will take you to calendar). ... Lothar Wolfgang Nordheim (November 7, 1899, München - October 5, 1985, La Jolla, California, USA) was a German-born Jewish theoretical physicist. ... Eugene Wigner Eugene Paul Wigner (Hungarian Wigner Pál Jenő) (November 17, 1902 – January 1, 1995) was a Hungarian physicist and mathematician who received the Nobel Prize in Physics in 1963 for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and... Hermann Weyl Hermann Weyl (November 9, 1885 – December 8, 1955) was a German mathematician. ... 1931 (MCMXXXI) was a common year starting on Thursday (link is to a full 1931 calendar). ...

Examples

In these examples, the underlying field of scalars is C, although similar definitions apply to the case in which the underlying field of scalars is R.

Euclidean spaces

Cⁿ with the inner product defined by

where the bar over a complex number denotes its complex conjugate. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...

Sequence spaces

Infinite-dimensional Hilbert spaces are central to the subject. If B is any set, the sequence space ℓ² (said "little ell two") over B is defined In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In functional analysis and related areas of mathematics, a sequence space is an important class of function space. ...

This space becomes a Hilbert space with the inner product

for all x and y in ℓ²(B). B does not have to be a countable set in this definition, although if B is not countable, the resulting Hilbert space is not separable. In a sense made more precise below, every Hilbert space is isomorphic to one of the form ℓ²(B) for a suitable set B. If B=N, the natural numbers, this space is simply called ℓ². In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...

Lebesgue spaces

These are function spaces associated to measure spaces (X, M, μ), where M is a σ-algebra of subsets of X and μ is a countably additive measure on M. Let L²_μ(X) be the space of complex-valued square-integrable measurable functions on X, modulo equality almost everywhere. Square integrable means the integral of the square of its absolute value is finite. Modulo equality almost everywhere means functions are identified if and only if they are equal outside of a set of measure 0. In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications, it is a topological space or a vector space or both. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ...

The inner product of functions f and g is here given by

One needs to show:

• That this integral indeed makes sense;
• The resulting space is complete.

These facts are easy to derive; see, for example, Section 42 of Halmos (1950).^[5] Note that the use of the Lebesgue integral ensures that the space will be complete. See L^p space for further discussion of this example. The integral can be interpreted as the area under a curve. ... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ...

Sobolev spaces

Sobolev spaces, denoted by H^s or W^s,2, are another example of Hilbert spaces, and are used often in the field of partial differential equations. In mathematics, a Sobolev space is a normed space of functions obtained by imposing on a function f and its weak derivatives up to some order k the condition of finite Lp norm, for given p ≥ 1. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ...

New Hilbert spaces from old

Two (or more) Hilbert spaces can be combined to produce another Hilbert space by taking either their direct sum or their tensor product. In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. ...

Applications

Hilbert spaces allow simple geometric concepts like projection and change of basis to be extended from finite dimensional to infinite dimensional spaces, in the first place, function spaces. Template:Unite See also projection (linear algebra). ... In linear algebra, we may consider some finite-dimensional vector space, which can have associated with it some basis with which we can work with respect to. ... In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications, it is a topological space or a vector space or both. ...

Other applications include:

• The theory of unitary group representations.
• The theory of square integrable stochastic processes.
• The Hilbert space theory of partial differential equations, in particular formulations of the Dirichlet problem.
• Spectral analysis of functions, including theories of wavelets.

One goal of Fourier analysis is to write a given function as a (possibly infinite) linear combination of given basis functions. This problem can be studied abstractly in Hilbert spaces: every Hilbert space has an orthonormal basis, and every element of the Hilbert space can be written in a unique way as a sum of multiples of these basis elements. The Fourier transform then corresponds to a change of basis. In mathematics, a unitary representation of a group G is a linear representation &#960; of G on a complex Hilbert space V such that &#960;(g) is a unitary operator for every g &#8712; G. The general theory is well-developed in case G is a locally compact (Hausdorff) topological... In the mathematics of probability, a stochastic process is a random function. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... In mathematics, Dirichlet problems are a class of partial differential equation (PDE) problems which ask you to solve for the values of a function in a region given the value of the function on the boundary of that region. ... In mathematics, wavelets, wavelet analysis, and the wavelet transform refers to the representation of a signal in terms of a finite length or fast decaying oscillating waveform (known as the mother wavelet). ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... In mathematics, an orthonormal basis of an inner product space V(i. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...

Orthonormal bases

A key role in the theory is played by the notion of orthonormal basis of a Hilbert space H: a family {e_k}_{k ∈ B} of H satisfying the conditions: In mathematics, an orthonormal basis of an inner product space V(i. ...

1. Orthogonality: Every two different elements of B are orthogonal: <e_k, e_j> = 0 for all k, j in B with k ≠ j.
2. Normalization: Every element of the family has norm 1: ||e_k|| = 1 for all k in B
3. Completeness: The linear span of B is dense in H.

A system of vectors satisfying the first two conditions basis is called an orthonormal system or an orthonormal sequence (if B is countable). It can be proved that such a system is always linearly independent. Completeness of an orthonormal system of vectors of a Hilbert space can be equivalently restated as: In the mathematical subfield of linear algebra, the linear span, also called the linear hull, of a set of vectors in a vector space is the intersection of all subspaces containing that set. ... In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if, intuitively, any point in X can be well-approximated by points in A. Formally, A is dense in X if for any point x in X, any neighborhood of... In mathematics, a countable set is a set with the same cardinality (i. ... In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ...

if for all and some then

Examples of orthonormal bases include:

• the set {(1,0,0),(0,1,0),(0,0,1)} forms an orthonormal basis of R³ with the dot product
• the sequence {f_n : n ∈ Z} with f_n(x) = exp(2πinx) forms an orthonormal basis of the complex space L²([0,1])
• the family {e_b : b ∈ B} with e_b(c) = 1 if b=c and 0 otherwise forms an orthonormal basis of l²(B).

Note that in the infinite-dimensional case, an orthonormal basis will not be a basis in the sense of linear algebra; to distinguish the two, the latter basis is also called a Hamel basis. That the span of the basis vectors is dense means that every vector in the space can be written as the limit of an infinite series and the orthogonality implies that this decomposition is unique. The exponential function is one of the most important functions in mathematics. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...

Using Zorn's lemma, one can show that every Hilbert space admits an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same cardinality. A Hilbert space is separable if and only if it admits a countable orthonormal basis. Zorns lemma, also known as the Kuratowski-Zorn lemma, is a proposition of set theory that states: Every non-empty partially ordered set in which every chain (i. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. ... In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space. ... In mathematics the term countable set is used to describe the size of a set, e. ...

Since all infinite-dimensional separable Hilbert spaces are isomorphic, and since almost all Hilbert spaces used in physics are separable, when physicists talk about the Hilbert space they mean any separable one. Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the fundamental laws of the universe. ...

If {e_k}_{k ∈ B} is an orthonormal basis of H, then every element x of H may be written as

Even if B is uncountable, only countably many terms in this sum will be non-zero, and the expression is therefore well-defined. This sum is also called the Fourier expansion of x.

If {e_k}_{k ∈ B} is an orthonormal basis of H, then H is isomorphic to l²(B) in the following sense: there exists a bijective linear map Φ : H → l²(B) such that In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...

for all x and y in H.

Orthogonal complements and projections

If S is a subset of a Hilbert space H, the set of vectors orthogonal to S is defined by

S^perp is a closed subspace of H and so forms itself a Hilbert space. If V is a closed subspace of H, then V^perp is called the orthogonal complement of V. In fact, every x in H can then be written uniquely as x = v + w, with v in V and w in V^perp. Therefore, H is the internal Hilbert direct sum of V and V^perp. The linear operator P_V : H → H which maps x to v is called the orthogonal projection onto V. In topology and related branches of mathematics, a closed set is a set whose complement is open. ...

Theorem. The orthogonal projection P_V is a self-adjoint linear operator on H of norm ≤ 1 with the property P_V² = P_V. Moreover, any self-adjoint linear operator E such that E² = E is of the form P_V, where V is the range of E. For every x in H, P_V(x) is the unique element v of V which minimizes the distance ||x - v||.

This provides the geometrical interpretation of P_V(x): it is the best approximation to x by elements of V.

Reflexivity

An important property of any Hilbert space is its reflexivity. In fact, more is true: one has a complete and convenient description of its dual space (the space of all continuous linear functions from the space H into the base field), which is itself a Hilbert space. Indeed, the Riesz representation theorem states that to every element φ of the dual H' there exists one and only one u in H such that This page concerns the reflexivity of a Banach space. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... There are several well-known theorems in functional analysis known as the Riesz representation theorem. ...

for all x in H and the association φ ↔ u provides an antilinear isomorphism between H and H'. This correspondence is exploited by the bra-ket notation popular in physics. Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ... Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the fundamental laws of the universe. ...

Bounded operators

For a Hilbert space H, the continuous linear operators A : H → H are of particular interest. Such a continuous operator is bounded in the sense that it maps bounded sets to bounded sets. This allows to define its norm as In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. ... In mathematics, the operator norm is a means to measure the size of certain linear operators. ...

The sum and the composition of two continuous linear operators is again continuous and linear. For y in H, the map that sends x to <y, Ax> is linear and continuous, and according to the Riesz representation theorem can therefore be represented in the form There are several well-known theorems in functional analysis known as the Riesz representation theorem. ...

This defines another continuous linear operator A^* : H → H, the adjoint of A. In mathematics, specifically in functional analysis, each linear operator on a Hilbert space has a corresponding adjoint operator. ...

The set L(H) of all continuous linear operators on H, together with the addition and composition operations, the norm and the adjoint operation, forms a C^*-algebra; in fact, this is the motivating prototype and most important example of a C^*-algebra. C*-algebras are an important area of research in functional analysis. ...

An element A of L(H) is called self-adjoint or Hermitian if A^* = A. These operators share many features of the real numbers and are sometimes seen as generalizations of them. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

An element U of L(H) is called unitary if U is invertible and its inverse is given by U^*. This can also be expressed by requiring that <Ux, Uy> = <x, y> for all x and y in H. The unitary operators form a group under composition, which can be viewed as the automorphism group of H. In functional analysis, a unitary operator is a bounded linear operator U on a Hilbert space satisfying U*U=UU*=I where I is the identity operator. ... This picture illustrates how the hours in a clock form a group. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...

Unbounded operators

If a linear operator has a closed graph and is defined on all of a Hilbert space, then, by the closed graph theorem in Banach space theory, it is necessarily bounded. However, unbounded operators can be obtained by defining a linear map on a proper subspace of the Hilbert space. In mathematics, the closed graph theorem is a basic result in functional analysis which characterizes continuous linear operators between Banach spaces in terms of the operator graph. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... Screenshot (from SSCX Star Warzone). ...

In quantum physics, several interesting unbounded operators are defined on a dense subspace of Hilbert space. It is possible to define self-adjoint unbounded operators, and these play the role of the observables in the mathematical formulation of quantum mechanics. In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if, intuitively, any point in X can be well-approximated by points in A. Formally, A is dense in X if for any point x in X, any neighborhood of... On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ...

Examples of self-adjoint unbounded operator on the Hilbert space L²(R) are:

• A suitable extension of the differential operator

where i is the imaginary unit and f is a differentiable function of compact support.

• The multiplication by x operator:

These correspond to the momentum and position observables, respectively. Note that neither A nor B is defined on all of H, since in the case of A the derivative need not exist, and in the case of B the product function need not be square integrable. In both cases, the set of possible arguments form dense subspaces of L²(R). In classical mechanics, momentum (pl. ... Definition In quantum mechanics, the position operator corresponds to the position observable of a particle. ...

See also

• Harmonic analysis
• Hermitian operators
• Mathematical analysis
• Operator algebra
• Riesz representation theorem
• Rigged Hilbert space
• Reproducing kernel Hilbert space
• Topologies on the set of operators on a Hilbert space

Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ... Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ... In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space (such as a Banach space), which is typically required to be closed in a specified operator topology. ... There are several well-known theorems in functional analysis known as the Riesz representation theorem. ... In mathematics, a rigged Hilbert space is a construction designed to link the distribution (test function) and square-integrable aspects of functional analysis. ... In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space is a function space in which pointwise evaluation is a continuous linear functional. ... In mathematics, the requirements of functional analysis mean there are several standard topologies which are given to the set of bounded linear operators on a Hilbert space. ...

Notes and references

1. ^ Kolmogorov, Andrey; S. V. Fomin (1970). Introductory Real Analysis, Revised English edition, trans. by Richard A. Silverman (1975), Dover Press. ISBN 0-486-61226-0. 
2. ^ Von Neumann, John (1929). "Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren". Mathematische Annalen 102: 49–131. 
3. ^ Hilbert, David; Lothar Nordheim and John von Neumann (1927). "Über die Grundlagen der Quantenmechanik". Mathematische Annalen 98: 1–30. 
4. ^ Weyl, Hermann (1931). The Theory of Groups and Quantum Mechanics, English edition (1950), Dover Press. ISBN 0-486-60269-9. 
5. ^ Halmos, Paul (1950). Measure Theory. D. van Nostrand Co. 

• Jean Dieudonné, Foundations of Modern Analysis, Academic Press, 1960.
• B.M. Levitan, "Hilbert space" SpringerLink Encyclopaedia of Mathematics (2001)