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Hilbert space hilbert space is a real or complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of finite-dimensional Euclidean space to possibly infinite-dimensional settings. The inner product extends the familiar dot product, making it possible to define length, angle, and orthogonality, while completeness guarantees that limits of Cauchy sequences remain inside the space. Hilbert space is a real or complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of finite-dimensional Euclidean space to possibly infinite-dimensional settings. Hilbert spaces were developed in the early 20th century through the work of David Hilbert, Erhard Schmidt, and Frigyes Riesz, and later placed in an abstract setting by Biography:John von Neumann.

Development

Hilbert spaces were developed in the early 20th century through the work of David Hilbert, Erhard Schmidt, and Frigyes Riesz, and later placed in an abstract setting by Biography:John von Neumann. They became central in functional analysis, Fourier analysis, the theory of partial differential equations, and the mathematical formulation of quantum mechanics.^[1]

A Hilbert space is in particular a Banach space, but with additional geometric structure coming from the inner product. This makes possible such notions as orthogonal projection, orthonormal bases, and Fourier expansion, all of which extend familiar geometric ideas from finite-dimensional vector spaces to infinite-dimensional ones.^[2]

Definition

Let $H$ be a complex vector space. An inner product on $H$ is a function assigning to each pair $x, y \in H$ a complex number $⟨ x, y ⟩$ such that:^[3]

$⟨ y, x ⟩ = \overline{⟨ x, y ⟩}$ (conjugate symmetry),
$⟨ a x_{1} + b x_{2}, y ⟩ = a ⟨ x_{1}, y ⟩ + b ⟨ x_{2}, y ⟩$ (linearity in the first argument),
$⟨ x, x ⟩ \geq 0$ , with equality if and only if $x = 0$ (positive definiteness).

It follows that the inner product is conjugate-linear in the second argument: $⟨ x, a y_{1} + b y_{2} ⟩ = \overline{a} ⟨ x, y_{1} ⟩ + \overline{b} ⟨ x, y_{2} ⟩ .$

Inner product

The inner product induces a norm $‖ x ‖ = \sqrt{⟨ x, x ⟩}$ and hence a distance function $d (x, y) = ‖ x - y ‖ .$ This turns $H$ into a metric space.^[3]

A Hilbert space is an inner product space that is complete with respect to this norm, meaning that every Cauchy sequence converges to an element of the space.^[4]

Basic geometric properties

The norm and inner product satisfy the Cauchy–Schwarz inequality $| ⟨ x, y ⟩ | \leq ‖ x ‖ ‖ y ‖,$ ^[5]

If $u$ and $v$ are orthogonal, then the Pythagorean theorem holds: $‖ u + v ‖^{2} = ‖ u ‖^{2} + ‖ v ‖^{2} .$ More generally, Hilbert spaces satisfy the parallelogram law: $‖ u + v ‖^{2} + ‖ u - v ‖^{2} = 2 (‖ u ‖^{2} + ‖ v ‖^{2}) .$ ^[6]

Important

One of the most important consequences of completeness is the existence of orthogonal projection onto closed subspaces: if $V$ is a closed linear subspace of a Hilbert space $H$ , then every element $x \in H$ can be written uniquely as $x = v + w$ with $v \in V$ and $w \in V^{⊥}$ .^[7]

Standard examples

A basic finite-dimensional example is $𝐑^{3}$ with the usual dot product $(\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}) \cdot (\begin{matrix} y_{1} \\ y_{2} \\ y_{3} \end{matrix}) = x_{1} y_{1} + x_{2} y_{2} + x_{3} y_{3} .$ ^[3]

Every finite-dimensional inner product space is automatically complete, and hence is a Hilbert space.^[3]

A fundamental infinite-dimensional example is the sequence space $ℓ^{2}$ , consisting of all sequences $(z_{1}, z_{2}, \dots)$ such that $\sum_{n = 1}^{\infty} | z_{n} |^{2} < \infty .$ Its inner product is $⟨ z, w ⟩ = \sum_{n = 1}^{\infty} z_{n} \overline{w_{n}} .$ ^[8]

Major class

Another major class is formed by Lebesgue spaces $L^{2} (X, μ)$ , where $\int_{X} | f |^{2} d μ < \infty .$ The inner product is $⟨ f, g ⟩ = \int_{X} f (t) \overline{g (t)} d μ (t) .$ ^[9]

Role in analysis and physics

Hilbert spaces provide the natural setting for Fourier series and Fourier transformss, since orthonormal bases allow functions to be expanded into convergent series of coefficients.^[10]

They are equally central in the weak formulation of partial differential equations, especially through Sobolev spacess and the Lax–Milgram theorem.^[11]

In modern quantum mechanics, pure states are represented by unit vectors in a complex Hilbert space, and observables by self-adjoint operators acting on that space.^[12]

History

The theory emerged from work on integral equations and orthogonal expansions. David Hilbert and Erhard Schmidt studied integral operators and eigenfunction expansions, while Frigyes Riesz and Ernst Otto Fischer proved the completeness of $L^{2}$ spaces, establishing one of the first genuine infinite-dimensional Hilbert spaces.^[13]

The abstract concept was clarified by Biography:John von Neumann, who introduced the term Hilbert space and made it central to operator theory and quantum theory.^[1]

@@ Line 1: / Line 1: @@
 {{Short description|Complete inner product space used in quantum mechanics and analysis}}
 {{Quantum book backlink|Foundations}}
+{{Quantum article nav|previous=Physics:Quantum Postulates|previous label=Postulates|next=Physics:Quantum Observables and operators|next label=Observables and operators}}
 <div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">
@@ Line 10: / Line 11: @@
 <div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
-'''Hilbert space''' is a [[real number|real]] or [[complex number|complex]] [[inner product space]] that is also a [[complete metric space]] with respect to the metric induced by the inner product. It generalizes the notion of finite-dimensional [[Euclidean space]] to possibly infinite-dimensional settings. The inner product extends the familiar [[dot product]], making it possible to define length, angle, and orthogonality, while completeness guarantees that limits of Cauchy sequences remain inside the space.<ref name="Rudin">{{cite book|last=Rudin|first=Walter|title=Functional Analysis|year=1991|isbn=9780070542365}}</ref>
+'''Hilbert space''' hilbert space is a real or complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of finite-dimensional Euclidean space to possibly infinite-dimensional settings. The inner product extends the familiar dot product, making it possible to define length, angle, and orthogonality, while completeness guarantees that limits of Cauchy sequences remain inside the space. Hilbert space is a real or complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of finite-dimensional Euclidean space to possibly infinite-dimensional settings. Hilbert spaces were developed in the early 20th century through the work of David Hilbert, Erhard Schmidt, and Frigyes Riesz, and later placed in an abstract setting by Biography:John von Neumann.
 </div>
@@ Line 19: / Line 20: @@
 </div>
-=Development=
+== Development ==
-Hilbert spaces were developed in the early 20th century through the work of [[Biography:David Hilbert|David Hilbert]], [[Biography:Erhard Schmidt|Erhard Schmidt]], and [[Biography:Frigyes Riesz|Frigyes Riesz]], and later placed in an abstract setting by [[Biography:John von Neumann]]. They became central in [[functional analysis]], [[Fourier analysis]], the theory of [[partial differential equation]]s, and the [[Physics:mathematical formulation of quantum mechanics|mathematical formulation of quantum mechanics]].<ref name="vonNeumann">{{cite book|last=von Neumann|first=John|title=Mathematical Foundations of Quantum Mechanics|year=1996|isbn=9780691028934}}</ref>
+Hilbert spaces were developed in the early 20th century through the work of David Hilbert, Erhard Schmidt, and Frigyes Riesz, and later placed in an abstract setting by Biography:John von Neumann. They became central in functional analysis, Fourier analysis, the theory of partial differential equations, and the mathematical formulation of quantum mechanics.<ref name="vonNeumann">{{cite book|last=von Neumann|first=John|title=Mathematical Foundations of Quantum Mechanics|year=1996|isbn=9780691028934}}</ref>
-A Hilbert space is in particular a [[Banach space]], but with additional geometric structure coming from the inner product. This makes possible such notions as orthogonal projection, orthonormal bases, and Fourier expansion, all of which extend familiar geometric ideas from finite-dimensional vector spaces to infinite-dimensional ones.<ref name="Halmos">{{cite book|last=Halmos|first=Paul|title=Introduction to Hilbert Space|year=1957|isbn=9780486817330}}</ref>
+A Hilbert space is in particular a Banach space, but with additional geometric structure coming from the inner product. This makes possible such notions as orthogonal projection, orthonormal bases, and Fourier expansion, all of which extend familiar geometric ideas from finite-dimensional vector spaces to infinite-dimensional ones.<ref name="Halmos">{{cite book|last=Halmos|first=Paul|title=Introduction to Hilbert Space|year=1957|isbn=9780486817330}}</ref>
 == Definition ==
@@ Line 40: / Line 41: @@
 This turns <math>H</math> into a metric space.<ref name="Axler"/>
-A '''Hilbert space''' is an inner product space that is complete with respect to this norm, meaning that every [[Cauchy sequence]] converges to an element of the space.<ref>{{cite book|last=Roman|first=Steven|title=Advanced Linear Algebra|year=2008|isbn=9780387728285|page=327}}</ref>
+A '''Hilbert space''' is an inner product space that is complete with respect to this norm, meaning that every Cauchy sequence converges to an element of the space.<ref>{{cite book|last=Roman|first=Steven|title=Advanced Linear Algebra|year=2008|isbn=9780387728285|page=327}}</ref>
 == Basic geometric properties ==
-The norm and inner product satisfy the [[Cauchy–Schwarz inequality]]
+The norm and inner product satisfy the Cauchy–Schwarz inequality
 <math display="block">|\langle x,y\rangle| \le \|x\|\|y\|,</math><ref>{{cite book|last=Dieudonné|first=Jean|title=Foundations of Modern Analysis|year=1960|isbn=9780122155505}}</ref>
-If <math>u</math> and <math>v</math> are orthogonal, then the [[Pythagorean theorem]] holds:
+If <math>u</math> and <math>v</math> are orthogonal, then the Pythagorean theorem holds:
 <math display="block">\|u+v\|^2 = \|u\|^2+\|v\|^2.</math>
-More generally, Hilbert spaces satisfy the [[parallelogram law]]:
+More generally, Hilbert spaces satisfy the parallelogram law:
 <math display="block">\|u+v\|^2+\|u-v\|^2 = 2(\|u\|^2+\|v\|^2).</math><ref>{{cite book|last=Reed|first=Michael|author2=Simon, Barry|title=Methods of Modern Mathematical Physics, Vol. 1|year=1980|isbn=9780125850506}}</ref>
 ===Important===
@@ Line 68: / Line 69: @@
 <math display="block">\langle z,w\rangle = \sum_{n=1}^{\infty} z_n\overline{w_n}.</math><ref>{{cite book|last=Stein|first=Elias|author2=Shakarchi, Rami|title=Real Analysis|year=2005|isbn=9780691113869}}</ref>
 ===Major class===
-Another major class is formed by [[Lp space|Lebesgue spaces]] <math>L^2(X,\mu)</math>, where
+Another major class is formed by Lebesgue spaces <math>L^2(X,\mu)</math>, where
 <math display="block">\int_X |f|^2\,\mathrm{d}\mu < \infty.</math>
 The inner product is
@@ Line 74: / Line 75: @@
 == Role in analysis and physics ==
-Hilbert spaces provide the natural setting for [[Fourier series]] and [[Fourier transform]]s, since orthonormal bases allow functions to be expanded into convergent series of coefficients.<ref>{{cite book|last=Folland|first=Gerald|title=Real Analysis|year=2009}}</ref>
+Hilbert spaces provide the natural setting for Fourier series and Fourier transformss, since orthonormal bases allow functions to be expanded into convergent series of coefficients.<ref>{{cite book|last=Folland|first=Gerald|title=Real Analysis|year=2009}}</ref>
-They are equally central in the weak formulation of [[partial differential equation]]s, especially through [[Sobolev space]]s and the [[Lax–Milgram theorem]].<ref>{{cite book|last=Brezis|first=Haim|title=Functional Analysis, Sobolev Spaces and PDEs|year=2010|doi=10.1007/978-0-387-70914-7}}</ref>
+They are equally central in the weak formulation of partial differential equations, especially through Sobolev spacess and the Lax–Milgram theorem.<ref>{{cite book|last=Brezis|first=Haim|title=Functional Analysis, Sobolev Spaces and PDEs|year=2010|doi=10.1007/978-0-387-70914-7}}</ref>
 In modern [[Physics:Quantum mechanics|quantum mechanics]], pure states are represented by unit vectors in a complex Hilbert space, and observables by self-adjoint operators acting on that space.<ref>{{cite book|last=Holevo|first=Alexander|title=Statistical Structure of Quantum Theory|year=2001|doi=10.1007/3-540-44998-1}}</ref>
 == History ==
-The theory emerged from work on [[integral equation]]s and orthogonal expansions. [[Biography:David Hilbert|David Hilbert]] and [[Biography:Erhard Schmidt|Erhard Schmidt]] studied integral operators and eigenfunction expansions, while [[Biography:Frigyes Riesz|Frigyes Riesz]] and [[Biography:Ernst Otto Fischer|Ernst Otto Fischer]] proved the completeness of <math>L^2</math> spaces, establishing one of the first genuine infinite-dimensional Hilbert spaces.<ref>{{cite book|last=Bourbaki|first=Nicolas|title=Topological Vector Spaces|year=1987|isbn=9780387136271}}</ref>
+The theory emerged from work on integral equations and orthogonal expansions. David Hilbert and Erhard Schmidt studied integral operators and eigenfunction expansions, while Frigyes Riesz and Ernst Otto Fischer proved the completeness of <math>L^2</math> spaces, establishing one of the first genuine infinite-dimensional Hilbert spaces.<ref>{{cite book|last=Bourbaki|first=Nicolas|title=Topological Vector Spaces|year=1987|isbn=9780387136271}}</ref>
-The abstract concept was clarified by [[Biography:John von Neumann]], who introduced the term ''Hilbert space'' and made it central to operator theory and quantum theory.<ref name="vonNeumann"/>
+The abstract concept was clarified by Biography:John von Neumann, who introduced the term ''Hilbert space'' and made it central to operator theory and quantum theory.<ref name="vonNeumann"/>
 =See also=

Physics:Quantum Hilbert space: Difference between revisions

Latest revision as of 12:15, 20 May 2026

Development

Definition

Inner product

Basic geometric properties

Important

Standard examples

Major class

Role in analysis and physics

History

See also

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