imported>WikiHarold: Replace old raw Quantum Collection backlink with B backlink template

2026-05-08T19:24:50Z

Replace old raw Quantum Collection backlink with B backlink template

← Older revision	Revision as of 19:24, 8 May 2026
(No difference)

imported>WikiHarold: Replace old raw Quantum Collection backlink with B backlink template

2026-05-08T19:24:50Z

Replace old raw Quantum Collection backlink with B backlink template

New page

{{Quantum book backlink|Foundations}}

'''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>
[[File:Standing_waves_on_a_string_yellow-1.gif|thumb|400px|style=background:#fff8dc;|The state of a vibrating string can be modeled as a point in a Hilbert space. Its decomposition into distinct overtones corresponds to orthogonal projection onto coordinate directions.]]

=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>

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 ==
Let <math>H</math> be a complex vector space. An '''inner product''' on <math>H</math> is a function assigning to each pair <math>x,y \in H</math> a complex number <math>\langle x,y\rangle</math> such that:<ref name="Axler">{{cite book|last=Axler|first=Sheldon|title=Linear Algebra Done Right|year=2024|isbn=9783031410253|pages=183–184}}</ref>

# <math>\langle y,x\rangle = \overline{\langle x,y\rangle}</math> (conjugate symmetry),
# <math>\langle ax_1+bx_2,y\rangle = a\langle x_1,y\rangle + b\langle x_2,y\rangle</math> (linearity in the first argument),
# <math>\langle x,x\rangle \ge 0</math>, with equality if and only if <math>x=0</math> (positive definiteness).

It follows that the inner product is conjugate-linear in the second argument:
<math display="block">\langle x, ay_1+by_2\rangle = \overline{a}\langle x,y_1\rangle + \overline{b}\langle x,y_2\rangle.</math>
===Inner product===
The inner product induces a norm
<math display="block">\|x\|=\sqrt{\langle x,x\rangle}</math>
and hence a distance function
<math display="block">d(x,y)=\|x-y\|.</math>
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>

== Basic geometric properties ==
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:
<math display="block">\|u+v\|^2 = \|u\|^2+\|v\|^2.</math>
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===
One of the most important consequences of completeness is the existence of orthogonal projection onto closed subspaces: if <math>V</math> is a closed linear subspace of a Hilbert space <math>H</math>, then every element <math>x \in H</math> can be written uniquely as
<math display="block">x=v+w</math>
with <math>v\in V</math> and <math>w\in V^\perp</math>.<ref name="Rudin"/>

== Standard examples ==
A basic finite-dimensional example is <math>\mathbf{R}^3</math> with the usual dot product
<math display="block">\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}\cdot
\begin{pmatrix}y_1\\y_2\\y_3\end{pmatrix}
=x_1y_1+x_2y_2+x_3y_3.</math><ref name="Axler"/>

Every finite-dimensional inner product space is automatically complete, and hence is a Hilbert space.<ref name="Axler"/>

A fundamental infinite-dimensional example is the sequence space <math>\ell^2</math>, consisting of all sequences <math>(z_1,z_2,\dots)</math> such that
<math display="block">\sum_{n=1}^{\infty}|z_n|^2 < \infty.</math>
Its inner product is
<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
<math display="block">\int_X |f|^2\,\mathrm{d}\mu < \infty.</math>
The inner product is
<math display="block">\langle f,g\rangle=\int_X f(t)\overline{g(t)}\,\mathrm{d}\mu(t).</math><ref>{{cite book|last=Halmos|first=Paul|title=Measure Theory|year=1950|doi=10.1007/978-1-4684-9440-2}}</ref>

== 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>

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>

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 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=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

= References =
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Physics:Quantum Hilbert space|1}}

Physics:Quantum Hilbert space - Revision history

imported>WikiHarold: Replace old raw Quantum Collection backlink with B backlink template

imported>WikiHarold: Replace old raw Quantum Collection backlink with B backlink template