The mathematical concept of a Hilbert space, named after David Hilbert, generalizes

8 min readMay 29, 2021

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is a vector space equipped with an inner product, an operation that allows lengths and angles to be defined. Furthermore, Hilbert spaces are complete, which means that there are enough limits in the space to allow the techniques of calculus to be used.

Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions.

Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a subspace (the analog of “dropping the altitude” of a triangle) plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to a set of coordinate axes (an orthonormal basis), in analogy with Cartesian coordinates in the plane. When that set of axes is countably infinite, the Hilbert space can also be usefully thought of in terms of the space of infinite sequences that are square-summable. The latter space is often in the older literature referred to as the Hilbert space. 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 a sense that is made precise by the study of their spectrum.

Contents
1 Definition and illustration
1.1 Motivating example: Euclidean vector space
1.2 Definition
1.3 Second example: sequence spaces
2 History
3 Examples
3.1 Lebesgue spaces
3.2 Sobolev spaces
3.3 Spaces of holomorphic functions
3.3.1 Hardy spaces
3.3.2 Bergman spaces
4 Applications
4.1 Sturm–Liouville theory
4.2 Partial differential equations
4.3 Ergodic theory
4.4 Fourier analysis
4.5 Quantum mechanics
4.6 Color perception
5 Properties
5.1 Pythagorean identity
5.2 Parallelogram identity and polarization
5.3 Best approximation
5.4 Duality
5.5 Weakly-convergent sequences
5.6 Banach space properties
6 Operators on Hilbert spaces
6.1 Bounded operators
6.2 Unbounded operators
7 Constructions
7.1 Direct sums
7.2 Tensor products
8 Orthonormal bases
8.1 Sequence spaces
8.2 Bessel’s inequality and Parseval’s formula
8.3 Hilbert dimension
8.4 Separable spaces
9 Orthogonal complements and projections
10 Spectral theory
11 In popular culture
12 See also
13 Remarks
14 Notes
15 References
16 External links
Definition and illustration
Motivating example: Euclidean vector space
One of the most familiar examples of a Hilbert space is the Euclidean vector space consisting of three-dimensional vectors, denoted by R3, and equipped with the dot product. The dot product takes two vectors x and y, and produces a real number x ⋅ y. If x and y are represented in Cartesian coordinates, then the dot product is defined by

{\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\cdot {\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}\,.}{\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\cdot {\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}\,.}
The dot product satisfies the properties:

It is symmetric in x and y: x ⋅ y = y ⋅ x.
It is linear in its first argument: (ax1 + bx2) ⋅ y = ax1 ⋅ y + bx2 ⋅ y for any scalars a, b, and vectors x1, x2, and y.
It is positive definite: for all vectors x, x ⋅ x ≥ 0 , with equality if and only if x = 0.
An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner product space. Every finite-dimensional inner product space is also a Hilbert space. The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length (or norm) of a vector, denoted ||x||, and to the angle θ between two vectors x and y by means of the formula

{\displaystyle \mathbf {x} \cdot \mathbf {y} =\left\|\mathbf {x} \right\|\left\|\mathbf {y} \right\|\,\cos \theta \,.}{\displaystyle \mathbf {x} \cdot \mathbf {y} =\left\|\mathbf {x} \right\|\left\|\mathbf {y} \right\|\,\cos \theta \,.}

Completeness means that if a particle moves along the broken path (in blue) travelling a finite total distance, then the particle has a well-defined net displacement (in orange).
Multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist. A mathematical series

{\displaystyle \sum _{n=0}^{\infty }\mathbf {x} _{n}}\sum _{n=0}^{\infty }\mathbf {x} _{n}
consisting of vectors in R3 is absolutely convergent provided that the sum of the lengths converges as an ordinary series of real numbers:[1]

{\displaystyle \sum _{k=0}^{\infty }\|\mathbf {x} _{k}\|<\infty \,.}{\displaystyle \sum _{k=0}^{\infty }\|\mathbf {x} _{k}\|<\infty \,.}
Just as with a series of scalars, a series of vectors that converges absolutely also converges to some limit vector L in the Euclidean space, in the sense that

{\displaystyle \left\|\mathbf {L} -\sum _{k=0}^{N}\mathbf {x} _{k}\right\|\to 0\quad {\text{as }}N\to \infty \,.}{\displaystyle \left\|\mathbf {L} -\sum _{k=0}^{N}\mathbf {x} _{k}\right\|\to 0\quad {\text{as }}N\to \infty \,.}
This property expresses the completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense.

Hilbert spaces are often taken over the complex numbers. The complex plane denoted by C is equipped with a notion of magnitude, the complex modulus |z| which is defined as the square root of the product of z with its complex conjugate:

{\displaystyle |z|^{2}=z{\overline {z}}\,.}{\displaystyle |z|^{2}=z{\overline {z}}\,.}
If z = x + iy is a decomposition of z into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length:

{\displaystyle |z|={\sqrt {x^{2}+y^{2}}}\,.}{\displaystyle |z|={\sqrt {x^{2}+y^{2}}}\,.}
The inner product of a pair of complex numbers z and w is the product of z with the complex conjugate of w:

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The mathematical concept of a Hilbert space, named after David Hilbert, generalizes

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