Hilbert space

For the Hilbert space-filling curve, see Hilbert curve. A Hilbert curve (also known as a Hilbert space-filling curve) is a continuous Fractal Space-filling curve first described by the German mathematician

This article assumes some familiarity with analytic geometry and the concept of a limit. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" The article on vector spaces contains useful background, and the article on functional analysis is closely related. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added For functional analysis as used in psychology see the Functional analysis (psychology article

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 two-dimensional plane and three-dimensional space to infinite-dimensional spaces. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most 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. In Mathematics, an inner product space is a Vector space with the additional Structure of 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 Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close"

Hilbert spaces arise naturally and frequently in mathematics, physics, and engineering, typically as infinite-dimensional function spaces. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Engineering is the Discipline and Profession of applying technical and scientific Knowledge and In Mathematics, a function space is a set of functions of a given kind from a set X to a set Y. They are indispensable tools in the theories of partial differential equations, quantum mechanics, and signal processing. In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of Quantum mechanics. Signal processing is the analysis interpretation and manipulation of signals Signals of interest include sound, images, biological signals such as 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. For functional analysis as used in psychology see the Functional analysis (psychology article

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. In Mathematics, an orthonormal basis of an Inner product space V (i When that basis is countably infinite, this means that the Hilbert space can also usefully be thought of in terms of infinite sequences that are square-summable. In Mathematics, a sequence is an ordered list of objects (or events In Mathematics, the Lp and ℓp spaces are spaces of p-power integrable functions, and corresponding 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, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that

Motivation

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. Distance is a numerical description of how far apart objects are In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R In Mathematics, a bilinear map is a function of two arguments that is linear in each In Mathematics, the real numbers may be described informally in several different ways Many problems from analytic geometry can be reworded and solved using the dot product, for example, "When are two lines orthogonal?" or "Which point on a given plane is closest to the origin?"

One of the insights of modern mathematics is that ideas from Euclidean geometry may be applied to problems which do not necessarily arise out of geometry. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry In Mathematics, two Vectors are orthogonal if they are Perpendicular, i 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 Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law In addition, the technical requirement of completeness ensures that certain limits exist. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" 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. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product.

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 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 particular, this principle applies to solving linear differential and integral equations, and especially eigenvalue problems. In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i In Mathematics, an integral equation is an equation in which an unknown function appears under an Integral sign In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes 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. Jean Baptiste Joseph Fourier ( March 21, 1768 &ndash May 16, 1830) was a French Mathematician and Physicist The heat equation is an important Partial differential equation which describes the distribution of Heat (or variation in temperature in a given region over time In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics Similar considerations apply to other equations of mathematical physics, notably, the wave equation and Helmholtz equation. The wave equation is an important second-order linear Partial differential equation that describes the propagation of a variety of Waves such as Sound waves The Helmholtz equation, named for Hermann von Helmholtz, is the Elliptic partial differential equation (\nabla^2 + k^2 A = 0

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. In Mathematics a Topological space is called separable if it contains a countable dense subset that is there exists a sequence \{ x_n

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. In Mathematics, an orthonormal basis of an Inner product space V (i 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.

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. In Linear algebra, a basis for a Vector space of dimension n is a sequence of n vectors &alpha1. In Mathematics, a function space is a set of functions of a given kind from a set X to a set Y.

Other applications include:

The theory of unitary group representations. In Mathematics, a unitary representation of a group G is a Linear representation π of G on a complex Hilbert space
The theory of square integrable stochastic processes. A stochastic process, or sometimes random process, is the counterpart to a deterministic process (or Deterministic system) in Probability theory.
The Hilbert space theory of partial differential equations, in particular formulations of the Dirichlet problem. In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i In Mathematics, a Dirichlet problem is the problem of finding a function which solves a specified Partial differential equation (PDE in the interior of
Spectral analysis of functions, including theories of wavelets. A wavelet is a mathematical function used to divide a given function or continuous-time signal into different frequency components and study each component with a resolution

One goal of Fourier analysis is to write a given function as a (possibly infinite) linear combination of given basis functions. In mathematics Fourier analysis is a subject area which grew out of the study of Fourier series In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics 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. In Mathematics, an orthonormal basis of an Inner product space V (i The Fourier transform then corresponds to a change of basis. This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and

History

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. Jean Baptiste Joseph Fourier ( March 21, 1768 &ndash May 16, 1830) was a French Mathematician and Physicist Friedrich Wilhelm Bessel (22 July 1784 &ndash 17 March 1846 was a German Mathematician, Astronomer, and systematizer of the Bessel functions Marc-Antoine Parseval des Chênes ( April 27, 1755 &ndash August 16, 1836) was a French Mathematician, most famous 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. In Mathematics, a trigonometric series is any series of the form \frac{1}{2}A_{o}+\displaystyle\sum_{n=1}^{\infty}(A_{n} \cos{nx} + B_{n} Further basic results were proved in early 20th century, for example, the Riesz representation theorem of Maurice Frechet and Frigyes Riesz from 1907. There are several well-known theorems in Functional analysis known as the Riesz representation theorem. Maurice Fréchet ( September 2, 1878 – June 4, 1973) was a French Mathematician. Frigyes Riesz ( January 22, 1880 &ndash February 28, 1956) was a Mathematician who was born in Győr, 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. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most In Mathematics, an integral equation is an equation in which an unknown function appears under an Integral sign ^[1] 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. For functional analysis as used in psychology see the Functional analysis (psychology article

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. The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description 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. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that The framework of Quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications 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. In Mathematics, a unitary representation of a group G is a Linear representation π of G on a complex Hilbert space In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of 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. Dynamical systems theory is an area of Applied mathematics used to describe the behavior of complex Dynamical systems usually by employing Differential Ergodic theory is a branch of Mathematics that studies Dynamical systems with an Invariant measure and related problems

John von Neumann coined the term abstract Hilbert space in his famous work on unbounded Hermitian operators (von Neumann 1929). In Mathematics, on a finite-dimensional Inner product space, a self-adjoint operator is one that is its own adjoint, or equivalently one whose matrix 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 in (Hilbert, Nordheim & von Neumann 1927), and continued in his work with Eugene Wigner. Eugene Paul "EP" Wigner ( Hungarian Wigner Pál Jenő) ( November 17, 1902 &ndash January 1, 1995) was a The name "Hilbert space" was soon adopted by others, for example by Hermann Weyl in his book on quantum mechanics and the theory of groups (Weyl 1931).

Definition and examples

A Hilbert space is a real or complex inner product space that is complete under the norm defined by the inner product $\langle\cdot,\cdot\rangle$ by^[2]

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

Some authors use slightly different definitions. 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 In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has For example, (Kolmogorov & Fomin 1970) define a Hilbert space as above but restrict the definition to separable infinite-dimensional spaces. In Mathematics a Topological space is called separable if it contains a countable dense subset that is there exists a sequence \{ x_n A separable, infinite-dimensional Hilbert space is unique up to isomorphism; it is denoted by ℓ²(N), or simply ℓ². Older books and papers sometimes call a Hilbert space a unitary space or a linear space with an inner product, but this terminology has fallen out of use.

In the examples of Hilbert spaces given below, the underlying field of scalars is the complex numbers C, although similar definitions apply to the case in which the underlying field of scalars is the real numbers R.

Euclidean spaces

Every finite-dimensional inner product space is also a Hilbert space. For example, Cⁿ with the inner product defined by

$\langle x, y \rangle = \sum_{k=1}^n x_k\overline{y_k}$

where the bar over a complex number denotes its complex conjugate. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the complex conjugate of a Complex number is given by changing the sign of the Imaginary part.

Sequence spaces

Given a set B, the sequence space ℓ² (said "little ell two") over B is defined

$\ell^2(B) =\big\{ x : B \xrightarrow{x} \mathbb{C} \text{ and } \sum_{b \in B} \left|x \left(b\right)\right|^2 < \infty \big\}.$

This space becomes a Hilbert space with the inner product

$\langle x, y \rangle = \sum_{b \in B} x(b)\overline{y(b)}$

for all x and y in ℓ²(B). In Functional analysis and related areas of Mathematics, a sequence space is a Vector space whose elements are infinite Sequences of Complex 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 Mathematics a Topological space is called separable if it contains a countable dense subset that is there exists a sequence \{ x_n Every Hilbert space is isomorphic to one of the form ℓ²(B) for a suitable set B. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective If B=N, the natural numbers, this space is separable and is simply called ℓ². In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an

Lebesgue spaces

Main article: Lp space

Lebesgue spaces are function spaces associated to measure spaces (X, M, μ), where X is a set, M is a σ-algebra of subsets of X, and μ is a countably additive measure on M. In Mathematics, the Lp and ℓp spaces are spaces of p-power integrable functions, and corresponding In Mathematics, a function space is a set of functions of a given kind from a set X to a set Y. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with In Mathematics, a σ-algebra (or sigma-algebra) ( Sigma is a Greek letter upper case Σ lower case σ over a set X is a nonempty In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with For example, Let L²_μ(X) be the space of those complex-valued measurable functions on X for which the Lebesgue integral of the square of the absolute value of the function is finite, and where functions are identified if and only if they differ only on a set of measure 0. In Mathematics, the Integral of a non-negative function can be regarded in the simplest case as the Area between the graph of In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. In Mathematics, a null set is a set that is negligible in some sense.

The inner product of functions f and g in L²_μ(X) is then defined as

$\langle f,g\rangle=\int_X f(t) \overline{g(t)} \ d \mu(t)$

This integral exists, and the resulting space is complete. See, for example, Halmos 1950, Section 42. The full Lebesgue integral is needed to ensure completeness, however, as not enough functions are Riemann integrable; see Hewitt & Stromberg 1965. In the branch of Mathematics known as Real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the Integral

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 Vector space of functions equipped with a norm that is a combination of ''Lp'' norms of the function In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i

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. The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector

Properties

Completeness

Completeness is the key to handling infinite-dimensional examples, such as function spaces, and is required, for instance, for the Riesz representation theorem to hold. There are several well-known theorems in Functional analysis known as the Riesz representation theorem. It is expressed using a form of the Cauchy criterion for sequences in H: a normed space H is complete if every Cauchy sequence converges with respect to this norm to an element in the space. The Cauchy convergence test is a method used to test infinite series for Convergence. 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 Cauchy sequence, named after Augustin Cauchy, is a Sequence whose elements become arbitrarily close to each other as the sequence In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close"

Parallelogram identity

By definition, every Hilbert space is also a Banach space. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis Furthermore, in every Hilbert space the following parallelogram identity holds:

$\|u+v\|^2+\|u-v\|^2=2(\|u\|^2+\|v\|^2).$

Conversely, every Banach space in which the parallelogram identity holds is a Hilbert space, and the inner product is uniquely determined by the norm. In Mathematics, the simplest form of the parallelogram law belongs to elementary Geometry.

Topology

As for 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. In Mathematics, a topological vector space is one of the basic structures investigated in Functional analysis. In Mathematics, a ball is the inside of a Sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions and for metric Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity.

Reflexivity

Every Hilbert space is reflexive, i. In Functional analysis, a Banach space is called reflexive if it satisfies a certain abstract property involving Dual spaces Reflexive spaces turn out to e. every Hilbert space can be naturally identified with its double dual. In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals 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. In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function 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

$\phi \left(x\right) = \langle u, x \rangle$

for all x in H and the association φ ↔ u provides an antilinear isomorphism between H and H'. There are several well-known theorems in Functional analysis known as the Riesz representation theorem. This correspondence is exploited by the bra-ket notation popular in physics. Bra-ket notation is a standard notation for describing Quantum states in the theory of Quantum mechanics composed of angle brackets (chevrons and Vertical Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion.

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:

Orthogonality: Every two different elements of B are orthogonal: <e_k, e_j> = 0 for all k, j in B with k ≠ j. In Mathematics, an orthonormal basis of an Inner product space V (i
Normalization: Every element of the family has norm 1: ||e_k|| = 1 for all k in B
Completeness: The linear span of B is dense in H. In the mathematical subfield of Linear algebra, the linear span, also called the linear hull, of a set of vectors in a Vector In Topology and related areas of Mathematics, a Subset A of a Topological space X is called dense (in X) if

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. 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 Completeness of an orthonormal system of vectors of a Hilbert space can be equivalently restated as:

if $\langle v, e_k\rangle=0$ for all $k\in B$ and some $v\in H,$ then $v=\mathbf{0}.$

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). The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x)

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. Linear algebra is the branch of Mathematics concerned with Basis vector redirects here For basis vector in the context of crystals see Crystal structure. 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.

Hilbert dimension

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, called the Hilbert dimension of the space. Zorn's lemma, also known as the Kuratowski-Zorn lemma, is a proposition of Set theory that states Every Partially ordered set in which This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. In detail, if {e_k}_{k ∈ B} is an orthonormal basis of H, then every element x of H may be written as

$x = \sum_{k \in B} \langle e_k , x \rangle e_k$

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

$\langle \Phi \left(x\right), \Phi\left(y\right) \rangle = \langle x, y \rangle$

for all x and y in H. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that The cardinal number of B is the Hilbert dimension of H. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English.

Separable spaces

A Hilbert space is separable if and only if it admits a countable 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 All infinite-dimensional separable Hilbert spaces are isomorphic to ℓ². In particular, since all infinite-dimensional separable Hilbert spaces are isomorphic, and since almost all Hilbert spaces used in physics are infinite-dimensional and separable, when physicists talk about "the Hilbert space" or just "Hilbert space", they mean any infinite-dimensional separable one. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion.

Orthogonal complements and projections

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

$S^\perp = \left\{ x \in H : \langle x, s \rangle = 0\ \forall s \in S \right\}$

S^⊥ is a closed subspace of H and so forms itself a Hilbert space. In Topology and related branches of Mathematics, a closed set is a set whose complement is open. If V is a closed subspace of H, then V^⊥ 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^⊥. Therefore, H is the internal Hilbert direct sum of V and V^⊥. The linear operator P_V : H → H which maps x to v is called the orthogonal projection onto V.

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.

Bounded operators

For a Hilbert space H, the continuous linear operators A : H → H are of particular interest. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that Such a continuous operator is bounded in the sense that it maps bounded sets to bounded sets. In Mathematical analysis and related areas of Mathematics, a set is called bounded, if it is in a certain sense of finite size This allows to define its norm as

$\lVert A \rVert = \sup \left\{\,\lVert Ax \rVert : \lVert x \rVert \leq 1\,\right\}.$

The sum and the composition of two continuous linear operators is again continuous and linear. In Mathematics, the operator norm is a means to measure the "size" of certain Linear operators Formally it is a norm defined on the space of 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

$\langle A^* y, x \rangle = \langle y, Ax \rangle.$

This defines another continuous linear operator A^* : H → H, the adjoint of A. There are several well-known theorems in Functional analysis known as the Riesz representation theorem. 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 (pronounced "C-star" are an important area of research in Functional analysis, a branch of Mathematics.

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 in several different ways

An element U of L(H) is called unitary if U is invertible and its inverse is given by U^*. In Functional analysis, a branch of Mathematics, a unitary operator is a Bounded linear operator U H → 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 Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element 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. In Mathematics, the closed graph theorem is a basic result in Functional analysis which characterizes Continuous linear operators between Banach spaces In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis However, unbounded operators can be obtained by defining a linear map on a proper subspace of the Hilbert space. Subspace may refer to;Mathematics Euclidean subspace, in linear algebra a set of vectors in n -dimensional Euclidean space that is closed under addition

In quantum physics, several interesting unbounded operators are defined on a dense subspace of Hilbert space. In Topology and related areas of Mathematics, a Subset A of a Topological space X is called dense (in X) if 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 Mathematics, on a finite-dimensional Inner product space, a self-adjoint operator is one that is its own adjoint, or equivalently one whose matrix

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

A suitable extension of the differential operator

$[A f](x) = i \frac{d}{dx} f(x), \quad$

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

The multiplication-by-x operator:

$[B f] (x) = xf(x).\quad$

These correspond to the momentum and position observables, respectively. In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In Quantum mechanics, the position operator corresponds to the position observable of a particle 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).

Notes

^ David Hilbert. Harmonic analysis is the branch of Mathematics that studies the representation of functions or signals as the superposition of basic Waves It investigates and generalizes In Mathematics, on a finite-dimensional Inner product space, a self-adjoint operator is one that is its own adjoint, or equivalently one whose matrix Hilbert C*-modules are Mathematical objects which generalise the notion of a Hilbert space (which itself is a generalisation of Euclidean space) in that Analysis has its beginnings in the rigorous formulation of Calculus. In Functional analysis, an operator algebra is an algebra of continuous Linear operators on a Topological vector space with the multiplication There are several well-known theorems in Functional analysis known as the Riesz representation theorem. In Mathematics, a rigged Hilbert space ( Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the In Functional analysis (a branch of Mathematics) a reproducing kernel Hilbert space is a Hilbert space of functions in which pointwise evaluation In Mathematics, the requirements of Functional analysis mean there are several standard topologies which are given to the algebra B ( H) of Encyclopædia Britannica (2007). Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Retrieved on 2007-09-08. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 70 - Roman forces under Titus sack Jerusalem. 1264 - The Statute of Kalisz
^ The mathematical material in this article can be found in any good textbook on functional analysis, such as (Dieudonné 1960).

References

Dieudonné, Jean (1960), Foundations of Modern Analysis, Academic Press
Halmos, Paul (1950), Measure Theory, D. Paul Richard Halmos ( March 3 1916 &mdash October 2 2006) was a Hungarian -born Jewish American Mathematician van Nostrand Co
Hewitt, Edwin & Stromberg, Karl (1965), Real and Abstract Analysis, Springer-Verlag
Hilbert, David; Nordheim, Lothar (Wolfgang) & von Neumann, John (1927), “Über die Grundlagen der Quantenmechanik”, Mathematische Annalen 98: 1–30, <http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?ht=VIEW&did=D27779>
Kolmogorov, Andrey & Fomin, Sergei V. (1970), Introductory Real Analysis (Revised English edition, trans. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most Lothar Wolfgang Nordheim ( November 7, 1899, München - October 5, 1985, La Jolla, California, USA was a Andrey Nikolaevich Kolmogorov (Андрей Николаевич Колмогоров ( April 25, 1903 - October 20, 1987) was a Soviet Sergei Vasilovich Fomin ( 9 December 1917 &ndash 17 August 1975) was a Russian Mathematician whoamong his other accomplishmentswas by Richard A. Silverman (1975) ed. ), Dover Press, ISBN 0-486-61226-0
B. M. Levitan (2001), “Hilbert space”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
von Neumann, John (1929), “Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren”, Mathematische Annalen 102: 49–131
Weyl, Hermann (1931), The Theory of Groups and Quantum Mechanics (English edition (1950) ed. The Encyclopaedia of Mathematics is a large reference work in Mathematics. Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician. ), Dover Press, ISBN 0-486-60269-9

Dictionary

-noun

(functional analysis) A generalized Euclidean space in which mathematical functions take the place of points; crucial to the understanding of quantum mechanics and other applications.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents