Physics:Quantum Hilbert space - Revision history

WikiHarold: Restore missing Quantum reference definitions

2026-05-24T00:31:11Z

Restore missing Quantum reference definitions

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	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		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>		<math display="block">x=v+w</math>
	with <math>v\in V</math> and <math>w\in V^\perp</math>.<ref name="Rudin"/>		with <math>v\in V</math> and <math>w\in V^\perp</math>.<ref name="Rudin">{{cite book\|last=Rudin\|first=Walter\|title=Functional Analysis\|year=1991\|isbn=9780070542365}}</ref>

	== Standard examples ==		== Standard examples ==

Maintenance script: Normalize quantum page header order

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	{{Quantum book backlink\|Foundations}}		{{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}}		{{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;">		<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">

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	{{Short description\|Complete inner product space used in quantum mechanics and analysis}}		{{Short description\|Complete inner product space used in quantum mechanics and analysis}}

	{{Quantum book backlink\|Foundations}}		{{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}}		{{Quantum article nav\|previous=Physics:Quantum Postulates\|previous label=Postulates\|next=Physics:Quantum Observables and operators\|next label=Observables and operators}}

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	{{Short description\|Complete inner product space used in quantum mechanics and analysis}}~~Book I~~		{{Short description\|Complete inner product space used in quantum mechanics and analysis}}

	{{Quantum book backlink\|Foundations}}		{{Quantum book backlink\|Foundations}}

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	~~~~{{Short description\|Complete inner product space used in quantum mechanics and analysis}}		{{Short description\|Complete inner product space used in quantum mechanics and analysis}}Book I

	{{Quantum book backlink\|Foundations}}		{{Quantum book backlink\|Foundations}}

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	~~~~{{Short description\|Complete inner product space used in quantum mechanics and analysis}}		{{Short description\|Complete inner product space used in quantum mechanics and analysis}}

	{{Quantum book backlink\|Foundations}}		{{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;">		<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">

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2026-05-20T09:02:20Z

Improve Book I intro quality

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	{{Short description\|Complete inner product space used in quantum mechanics and analysis}}		{{Short description\|Complete inner product space used in quantum mechanics and analysis}}

	{{Quantum book backlink\|Foundations}}		{{Quantum book backlink\|Foundations}}
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	<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">		<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 Book I topic in the Quantum Collection. 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 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>		</div>

Maintenance script: Clean missing Book I links and template output

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Clean missing Book I links and template output

Maintenance script: Clean Book I red links, intro, and image slots

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Clean Book I red links, intro, and image slots

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	{{Short description\|Complete inner product space used in quantum mechanics and analysis}}		{{Short description\|Complete inner product space used in quantum mechanics and analysis}}

	{{Quantum book backlink\|Foundations}}		{{Quantum book backlink\|Foundations}}
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	<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">		<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''' is a Book I topic in the Quantum Collection. 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.
	</div>		</div>

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	== 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 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>		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 ==		== Definition ==
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	This turns <math>H</math> into a metric space.<ref name="Axler"/>		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 ==		== 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>		<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>		<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]]:
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	<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>		<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===		===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>		<math display="block">\int_X \|f\|^2\,\mathrm{d}\mu < \infty.</math>
	The inner product is		The inner product is
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	== Role in analysis and physics ==		== 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 equation]]s, 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>		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 ==		== 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 equation]]s 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=		=See also=

Harold: Clean Quantum header BOM and legacy backlink

2026-05-17T22:04:54Z

Clean Quantum header BOM and legacy backlink

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	{{Short description\|Complete inner product space used in quantum mechanics and analysis}}		{{Short description\|Complete inner product space used in quantum mechanics and analysis}}

	{{Quantum book backlink\|Foundations}}		{{Quantum book backlink\|Foundations}}

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	{{Short description\|Complete inner product space used in quantum mechanics and analysis}}		{{Short description\|Complete inner product space used in quantum mechanics and analysis}}

	{{Quantum book backlink\|Foundations}}		{{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}}		{{Quantum article nav\|previous=Physics:Quantum Postulates\|previous label=Postulates\|next=Physics:Quantum Observables and operators\|next label=Observables and operators}}

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	{{Short description\|Complete inner product space used in quantum mechanics and analysis}}		{{Short description\|Complete inner product space used in quantum mechanics and analysis}}

	{{Quantum book backlink\|Foundations}}		{{Quantum book backlink\|Foundations}}
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	== Development ==		== 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 ~~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>
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	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>		<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>		<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===		===Important===
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	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>		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 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>		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>		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 ==		== History ==
	The theory emerged from work on [[integral ~~equation]]s~~ 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 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"/>