# Hilbert space
A **Hilbert space** is a vector space over the real or complex numbers equipped with an inner product, a function that assigns a scalar \( (x, y) \) to each pair of vectors, with the additional property of completeness: every [Cauchy sequence](https://www.edgepedia.org/cauchy-sequence) of vectors converges to a vector in the space. It generalizes finite-dimensional [Euclidean geometry](https://www.edgepedia.org/euclidean-geometry), with its notions of length, angle, and perpendicularity, to infinite-dimensional settings, and it is the natural setting for the theory of linear operators used in quantum mechanics, signal processing, and the solution of partial differential equations.[1](https://encyclopediaofmath.org/wiki/Hilbert_space)
| Key fact | Detail |
|---|---|
| Definition | A complete vector space over \( \mathbb{R} \) or \( \mathbb{C} \) with an inner product \( (x, y) \) satisfying positivity, linearity, and conjugate symmetry axioms[1](https://encyclopediaofmath.org/wiki/Hilbert_space) |
| Norm and metric | \( \|x\| = (x, x)^{1/2} \); the distance \( \rho(x, y) = \|x - y\| \) makes the space a metric space[1](https://encyclopediaofmath.org/wiki/Hilbert_space) |
| Cauchy–Schwarz inequality | \( \|(x, y)\| \leq \|x\| \cdot \|y\| \) holds in every Hilbert space[1](https://encyclopediaofmath.org/wiki/Hilbert_space) |
| Standard examples | The sequence space \( \ell^2 \) of square-summable sequences and the function space \( L^2 \) of square-integrable functions[2](https://faculty.etsu.edu/gardnerr/Func/notes/HWG-5-3.pdf) • [3](https://www.nou.edu.ng/coursewarecontent/MTH412.pdf) |
| Classification | Every infinite-dimensional separable Hilbert space is isomorphic to \( \ell^2 \)[4](https://www.cfm.brown.edu/people/dobrush/am36/Mathematica/ch2/hilbert.html) |
| Duality | Every linear functional has the form \( f(x) = (x, x^*) \) for a unique \( x^* \in H \) with \( \|f\| = \|x^*\| \) (Riesz representation)[1](https://encyclopediaofmath.org/wiki/Hilbert_space) • [5](https://math.mit.edu/~kehle/files/Introduction_to_functional_analysis_18_102.pdf) |
| Origin | Concept formulated in the work of David Hilbert and Erhard Schmidt on integral equations; abstract definition given by von Neumann, Frigyes Riesz, and Marshall Stone[1](https://encyclopediaofmath.org/wiki/Hilbert_space) |
## Definition
The Encyclopedia of Mathematics states the definition through axioms on a function \( (x, y) \) defined on \( H \times H \): it is positive definite, \( (x, x) = 0 \) if and only if \( x = 0 \) and \( (x, x) \geq 0 \) otherwise; linear in the first argument; and conjugate symmetric, \( (x, y) = \overline{(y, x)} \). Completeness is stated as a convergence condition: if a sequence \( (x_n) \) satisfies \( \lim_{n, m \to \infty} (x_n - x_m, x_n - x_m) = 0 \), then some \( x \in H \) exists with \( (x - x_n, x - x_n) \to 0 \).[1](https://encyclopediaofmath.org/wiki/Hilbert_space) In modern language, the norm \( \|x\| = (x, x)^{1/2} \) turns \( H \) into a metric space via \( \rho(x, y) = \|x - y\| \), and completeness means every Cauchy sequence converges in that metric.[1](https://encyclopediaofmath.org/wiki/Hilbert_space)
Terminology varies among authors. Some definitions require infinite dimensionality; others call an incomplete inner product space a pre-Hilbert space and reserve Hilbert space for its completion. Under the common convention that drops the dimensionality condition, a Hilbert space is simply a complete pre-Hilbert space.[1](https://encyclopediaofmath.org/wiki/Hilbert_space) In a finite-dimensional vector space over the real or complex numbers, completeness is automatic once an inner product is specified, so the distinction only matters in infinite dimensions.[1](https://encyclopediaofmath.org/wiki/Hilbert_space) Any pre-Hilbert space can be completed to a Hilbert space.[1](https://encyclopediaofmath.org/wiki/Hilbert_space)
## Examples
**Sequence spaces.** The space \( \ell^2(\mathbb{R}) \) consists of real sequences \( (x_1, x_2, \ldots) \) for which \( \sum_{n=1}^{\infty} x_n^2 < \infty \), and \( \ell^2(\mathbb{C}) \) its complex counterpart with \( \sum_{n=1}^{\infty} |z_n|^2 < \infty \); both are Hilbert spaces.[2](https://faculty.etsu.edu/gardnerr/Func/notes/HWG-5-3.pdf) Completeness can be shown directly by coordinating the convergence of a Cauchy sequence of sequences.[6](https://maths-people.anu.edu.au/~john/Assets/Analysis2_2012.pdf)
**Function spaces.** The space \( L^2[a, b] \) of complex-valued square-integrable functions on an interval is a Hilbert space with an inner product given by an integral.[3](https://www.nou.edu.ng/coursewarecontent/MTH412.pdf) More generally, \( L^2(E) \) for a measurable set \( E \subset \mathbb{R}^d \) is a Hilbert space.[6](https://maths-people.anu.edu.au/~john/Assets/Analysis2_2012.pdf) [Lebesgue integration](https://www.edgepedia.org/lebesgue-integration) is what makes this space complete: von Neumann, in constructing a concrete model of the separable Hilbert space, specified that integration be defined in the manner of Lebesgue so that Cauchy sequences of square-integrable functions converge in the mean, and he took the elements to be equivalence classes of functions differing only on sets of measure zero.[7](https://plato.stanford.edu/entries/qt-nvd/)
**Classification.** Up to isometric isomorphism, meaning a one-to-one correspondence preserving the linear operations and the inner product, the infinite-dimensional separable case is a single object: any infinite-dimensional separable Hilbert space is isomorphic to \( \ell^2 \).[4](https://www.cfm.brown.edu/people/dobrush/am36/Mathematica/ch2/hilbert.html) • [1](https://encyclopediaofmath.org/wiki/Hilbert_space)
## Geometry and structure
The inner product supplies the geometry. The [Cauchy–Schwarz inequality](https://www.edgepedia.org/cauchy-schwarz-inequality), \( |(x, y)| \leq \|x\| \cdot \|y\| \), underlies the notions of angle and orthogonality.[1](https://encyclopediaofmath.org/wiki/Hilbert_space) Orthogonal projections behave as in finite dimensions: for a closed linear subspace \( P \) of a Hilbert space, each vector \( u \) has a unique decomposition into a component in \( P \) and one orthogonal to it.[8](https://math.mit.edu/~rbm/18-102-S14/Pre-Final.pdf) An orthogonal sequence \( \{\varphi_n\} \) in a separable Hilbert space is complete precisely when no nonzero vector is orthogonal to all of its members.[4](https://www.cfm.brown.edu/people/dobrush/am36/Mathematica/ch2/hilbert.html)
**Duality.** Linear functionals on a Hilbert space have an especially simple form. Every linear functional \( f \) on \( H \) can be written uniquely as \( f(x) = (x, x^*) \) with \( x^* \in H \) and \( \|f\| = \|x^*\| \); the dual space \( H^* \) is isometrically anti-isomorphic to \( H \).[1](https://encyclopediaofmath.org/wiki/Hilbert_space) The Riesz representation theorem states this with the map \( f \mapsto z_f \) an anti-linear isometric bijection satisfying \( f(x) = \langle z_f, x \rangle \) for all \( x \).[5](https://math.mit.edu/~kehle/files/Introduction_to_functional_analysis_18_102.pdf) In bra-ket notation, the correspondence sends a vector \( |\psi\rangle \in H \) to the functional \( \langle \psi| \in H^* \).[5](https://math.mit.edu/~kehle/files/Introduction_to_functional_analysis_18_102.pdf) Reflexivity follows, and with it two further facts: a Hilbert space is weakly sequentially complete, and a subset is relatively weakly compact if and only if it is bounded.[1](https://encyclopediaofmath.org/wiki/Hilbert_space)
## Operators and the spectral theorem
The main content of the theory of Hilbert spaces is the theory of linear operators on them.[1](https://encyclopediaofmath.org/wiki/Hilbert_space) The bounded operators \( \mathfrak{B}(H) \), with addition, scalar multiplication, composition, and the adjoint operation, form a Banach algebra with involution; the self-adjoint, unitary, and normal operators are the well-studied classes with special behavior relative to the inner product.[1](https://encyclopediaofmath.org/wiki/Hilbert_space) The spectral decomposition, which constructs any self-adjoint, unitary, or normal operator from orthogonal projections, is especially simple for separable Hilbert spaces and extends to unbounded self-adjoint and normal operators.[1](https://encyclopediaofmath.org/wiki/Hilbert_space) Hilbert himself found the spectral decomposition for bounded self-adjoint operators on \( \ell_2 \) and introduced the resolution of the identity; von Neumann later found the decomposition for unbounded self-adjoint operators, with intervening contributions by [Torsten Carleman](https://www.edgepedia.org/torsten-carleman) on symmetric integral operators.[1](https://encyclopediaofmath.org/wiki/Hilbert_space)
## History
The concept arose from the theory of integral equations. Between 1904 and 1906, [David Hilbert](https://www.edgepedia.org/david-hilbert) (1862–1943) studied what were later named integral equations of the second kind, and the concept of a Hilbert space was formulated in the works of Hilbert and [Erhard Schmidt](https://www.edgepedia.org/erhard-schmidt) on that subject.[1](https://encyclopediaofmath.org/wiki/Hilbert_space) • [9](https://www.researchgate.net/publication/228577231_Some_remarks_on_the_history_of_functional_analysis) The abstract definition was given by von Neumann, Frigyes Riesz, and Marshall Stone in their studies of Hermitian operators.[1](https://encyclopediaofmath.org/wiki/Hilbert_space) The function spaces introduced by Hilbert, Maurice Fréchet, and [Frigyes Riesz](https://www.edgepedia.org/frigyes-riesz) between 1904 and 1918 also served as the starting point for the theory of Banach spaces, the complete normed spaces that generalize Hilbert spaces by dropping the inner product.[10](https://encyclopediaofmath.org/wiki/Banach_space)
## Quantum mechanics
In the late 1920s, [John von Neumann](https://www.edgepedia.org/john-von-neumann) developed the separable Hilbert space formulation of quantum mechanics, which became the definitive one from the standpoint of mathematical rigor; his 1932 treatise summarized it.[7](https://plato.stanford.edu/entries/qt-nvd/) In this formalism, states of a physical system are normalized vectors in a complex, separable, complete inner product space, observables are self-adjoint operators, and the expected value of an observable \( A \) in a unit-vector state \( u \) is the inner product \( \langle Au, u \rangle \).[7](https://plato.stanford.edu/entries/qt-nvd/) • [11](https://plato.stanford.edu/archives/sum2016/entries/qt-quantlog/) Von Neumann imposed separability, having a countable basis, as an additional requirement in his definition, and he rejected Dirac's use of delta functions and the assumption that every self-adjoint operator can be put in diagonal form as mathematically unacceptable.[7](https://plato.stanford.edu/entries/qt-nvd/) The separable Hilbert space remains a crucial framework for quantum theory, with finite-dimensional examples such as the two-dimensional space describing a qubit.[7](https://plato.stanford.edu/entries/qt-nvd/) • [12](https://plato.stanford.edu/entries/qt-quantcomp/)
Extensions handle situations the plain space does not capture. The rigged Hilbert space, characterized by [Israel Gelfand](https://www.edgepedia.org/israel-gelfand) and Neumark Vilenkin in 1964, builds on [Laurent Schwartz](https://www.edgepedia.org/laurent-schwartz)'s theory of distributions and [Alexander Grothendieck](https://www.edgepedia.org/alexander-grothendieck)'s nuclear spaces and provides a rigorous way to handle observables such as position and momentum as Dirac presented them.[13](https://plato.stanford.edu/archives/win2011/entries/qt-nvd/) In quantum probability, each proposition is represented by a projection operator on a Hilbert space, forming a non-distributive orthocomplemented lattice in place of the Boolean algebra of classical events.[11](https://plato.stanford.edu/archives/sum2016/entries/qt-quantlog/)
## Other applications
The Hilbert space is one of the most commonly used mathematical frameworks in signal processing and the associated optimization, where tools such as the windowed [Fourier transform](https://www.edgepedia.org/fourier-transform) provide information about a signal simultaneously in time and frequency domains.[14](https://www.mdpi.com/2413-4155/4/4/40) Least-squares regression over Hilbert spaces, including reproducing kernel Hilbert spaces, is an active setting for statistical learning theory.[15](https://arxiv.org/abs/1803.04371)
## References
1. [Hilbert space, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Hilbert_space)
2. [The Space L2, ETSU functional analysis notes](https://faculty.etsu.edu/gardnerr/Func/notes/HWG-5-3.pdf)
3. [National Open University of Nigeria, MTH412 courseware](https://www.nou.edu.ng/coursewarecontent/MTH412.pdf)
4. [Mathematica tutorial, Part 4.2: Hilbert spaces, Brown University](https://www.cfm.brown.edu/people/dobrush/am36/Mathematica/ch2/hilbert.html)
5. [Introduction to Functional Analysis 18.102, MIT (K. Kehle)](https://math.mit.edu/~kehle/files/Introduction_to_functional_analysis_18_102.pdf)
6. [Analysis 2, 2012 Lecture Notes, Australian National University](https://maths-people.anu.edu.au/~john/Assets/Analysis2_2012.pdf)
7. [Quantum Theory and Mathematical Rigor, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/qt-nvd/)
8. [Questions 18.102 Final, Spring 2014, MIT](https://math.mit.edu/~rbm/18-102-S14/Pre-Final.pdf)
9. [Some remarks on the history of functional analysis, ResearchGate](https://www.researchgate.net/publication/228577231_Some_remarks_on_the_history_of_functional_analysis)
10. [Banach space, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Banach_space)
11. [Quantum Logic and Probability Theory, Stanford Encyclopedia of Philosophy (Summer 2016 edition)](https://plato.stanford.edu/archives/sum2016/entries/qt-quantlog/)
12. [Quantum Computing, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/qt-quantcomp/)
13. [Quantum Theory: von Neumann vs. Dirac, Stanford Encyclopedia of Philosophy (Winter 2011 edition)](https://plato.stanford.edu/archives/win2011/entries/qt-nvd/)
14. [A Concise Tutorial on Functional Analysis for Applications to Signal Processing, MDPI](https://www.mdpi.com/2413-4155/4/4/40)
15. [Optimal Rates of Sketched-regularized Algorithms for Least-Squares Regression over Hilbert Spaces, arXiv](https://arxiv.org/abs/1803.04371)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Analysis and mathematical models › Functional analysis*
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