Hilbert 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

$${\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\cdot {\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}\,.}$$

The dot product satisfies the properties:

1. It is symmetric in x and y: x ⋅ y = y ⋅ x.
2. It is linear in its first argument: (ax₁ + bx₂) ⋅ y = a(x₁ ⋅ y) + b(x₂ ⋅ y) for any scalars a, b, and vectors x₁, x₂, and y.
3. 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.[1] 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

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

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

$${\displaystyle \sum _{n=0}^{\infty }\mathbf {x} _{n}}$$

consisting of vectors in R³ is absolutely convergent provided that the sum of the lengths converges as an ordinary series of real numbers:[2]

$${\displaystyle \sum _{k=0}^{\infty }\|\mathbf {x} _{k}\|<\infty \,.}$$

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

$${\displaystyle \left\|\mathbf {L} -\sum _{k=0}^{N}\mathbf {x} _{k}\right\|\to 0\quad {\text{as }}N\to \infty \,.}$$

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:

$${\displaystyle |z|^{2}=z{\overline {z}}\,.}$$

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:

$${\displaystyle |z|={\sqrt {x^{2}+y^{2}}}\,.}$$

The inner product of a pair of complex numbers z and w is the product of z with the complex conjugate of w:

$${\displaystyle \langle z,w\rangle =z{\overline {w}}\,.}$$

This is complex-valued. The real part of ⟨z, w⟩ gives the usual two-dimensional Euclidean dot product.

A second example is the space C² whose elements are pairs of complex numbers z = (z₁, z₂). Then the inner product of z with another such vector w = (w₁, w₂) is given by

$${\displaystyle \langle z,w\rangle =z_{1}{\overline {w_{1}}}+z_{2}{\overline {w_{2}}}\,.}$$

The real part of ⟨z, w⟩ is then the two-dimensional Euclidean dot product. This inner product is Hermitian symmetric, which means that the result of interchanging z and w is the complex conjugate:

$${\displaystyle \langle w,z\rangle ={\overline {\langle z,w\rangle }}\,.}$$

A Hilbert space H is a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product.[3]

To say that H is a complex inner product space means that H is a complex vector space on which there is an inner product ${\displaystyle \langle x,y\rangle }$ associating a complex number to each pair of elements ${\displaystyle x,y}$ of H that satisfies the following properties:

1. The inner product is conjugate symmetric; that is, the inner product of a pair of elements is equal to the complex conjugate of the inner product of the swapped elements:
   $${\displaystyle \langle y,x\rangle ={\overline {\langle x,y\rangle }}\,.}$$

   
   Importantly, this implies that ${\displaystyle \langle x,x\rangle }$ is a real number.
2. The inner product is linear in its first[nb 1] argument. For all complex numbers ${\displaystyle a}$ and ${\displaystyle b,}$
   $${\displaystyle \langle ax_{1}+bx_{2},y\rangle =a\langle x_{1},y\rangle +b\langle x_{2},y\rangle \,.}$$

   
3. The inner product of an element with itself is positive definite:
   $${\displaystyle {\begin{alignedat}{4}\langle x,x\rangle >0&\quad {\text{ if }}x\neq 0,\\\langle x,x\rangle =0&\quad {\text{ if }}x=0\,.\end{alignedat}}}$$

   

It follows from properties 1 and 2 that a complex inner product is antilinear, also called conjugate linear, in its second argument, meaning that

$${\displaystyle \langle x,ay_{1}+by_{2}\rangle ={\bar {a}}\langle x,y_{1}\rangle +{\bar {b}}\langle x,y_{2}\rangle \,.}$$

A real inner product space is defined in the same way, except that H is a real vector space and the inner product takes real values. Such an inner product will be a bilinear map and ${\displaystyle (H,H,\langle \cdot ,\cdot \rangle )}$ will form a dual system.[4]

The norm is the real-valued function

$${\displaystyle \|x\|={\sqrt {\langle x,x\rangle }}\,,}$$

and the distance ${\displaystyle d}$ between two points ${\displaystyle x,y}$ in H is defined in terms of the norm by

$${\displaystyle d(x,y)=\|x-y\|={\sqrt {\langle x-y,x-y\rangle }}\,.}$$

That this function is a distance function means firstly that it is symmetric in ${\displaystyle x}$ and ${\displaystyle y,}$ secondly that the distance between ${\displaystyle x}$ and itself is zero, and otherwise the distance between ${\displaystyle x}$ and ${\displaystyle y}$ must be positive, and lastly that the triangle inequality holds, meaning that the length of one leg of a triangle xyz cannot exceed the sum of the lengths of the other two legs:

$${\displaystyle d(x,z)\leq d(x,y)+d(y,z)\,.}$$

This last property is ultimately a consequence of the more fundamental Cauchy–Schwarz inequality, which asserts

$${\displaystyle \left|\langle x,y\rangle \right|\leq \|x\|\|y\|}$$

with equality if and only if ${\displaystyle x}$ and ${\displaystyle y}$ are linearly dependent.

With a distance function defined in this way, any inner product space is a metric space, and sometimes is known as a Hausdorff pre-Hilbert space.[5] Any pre-Hilbert space that is additionally also a complete space is a Hilbert space.

The completeness of H is expressed using a form of the Cauchy criterion for sequences in H: a pre-Hilbert space H is complete if every Cauchy sequence converges with respect to this norm to an element in the space. Completeness can be characterized by the following equivalent condition: if a series of vectors

$${\displaystyle \sum _{k=0}^{\infty }u_{k}}$$

converges absolutely in the sense that

$${\displaystyle \sum _{k=0}^{\infty }\|u_{k}\|<\infty \,,}$$

then the series converges in H, in the sense that the partial sums converge to an element of H.

As a complete normed space, Hilbert spaces are by definition also Banach spaces. As such they are topological vector spaces, in which topological notions like the openness and closedness of subsets are well defined. Of special importance is the notion of a closed linear subspace of a Hilbert space that, with the inner product induced by restriction, is also complete (being a closed set in a complete metric space) and therefore a Hilbert space in its own right.

The sequence space l² consists of all infinite sequences z = (z₁, z₂, …) of complex numbers such that the following series converges:

$${\displaystyle \sum _{n=1}^{\infty }|z_{n}|^{2}}$$

The inner product on l² is defined by:

$${\displaystyle \langle \mathbf {z} ,\mathbf {w} \rangle =\sum _{n=1}^{\infty }z_{n}{\overline {w_{n}}}\,,}$$

This second series converges as a consequence of the Cauchy–Schwarz inequality and the convergence of the previous series.

Completeness of the space holds provided that whenever a series of elements from l² converges absolutely (in norm), then it converges to an element of l². The proof is basic in mathematical analysis, and permits mathematical series of elements of the space to be manipulated with the same ease as series of complex numbers (or vectors in a finite-dimensional Euclidean space).[6]

Lebesgue spaces are function spaces associated to measure spaces (X, M, μ), where X is a set, M is a σ-algebra of subsets of X, and μ is a countably additive measure on M. Let L²(X, μ) be the space of those complex-valued measurable functions on X for which the Lebesgue integral of the square of the absolute value of the function is finite, i.e., for a function f in L²(X, μ),

$${\displaystyle \int _{X}|f|^{2}\mathrm {d} \mu <\infty \,,}$$

and where functions are identified if and only if they differ only on a set of measure zero.

The inner product of functions f and g in L²(X, μ) is then defined as

$${\displaystyle \langle f,g\rangle =\int _{X}f(t){\overline {g(t)}}\,\mathrm {d} \mu (t)}$$

or

$${\displaystyle \langle f,g\rangle =\int _{X}{\overline {f(t)}}g(t)\,\mathrm {d} \mu (t)\,,}$$

where the second form (conjugation of the first element) is commonly found in the theoretical physics literature. For f and g in L², the integral exists because of the Cauchy–Schwarz inequality, and defines an inner product on the space. Equipped with this inner product, L² is in fact complete.[24] The Lebesgue integral is essential to ensure completeness: on domains of real numbers, for instance, not enough functions are Riemann integrable.[25]

The Lebesgue spaces appear in many natural settings. The spaces L²(R) and L²([0,1]) of square-integrable functions with respect to the Lebesgue measure on the real line and unit interval, respectively, are natural domains on which to define the Fourier transform and Fourier series. In other situations, the measure may be something other than the ordinary Lebesgue measure on the real line. For instance, if w is any positive measurable function, the space of all measurable functions f on the interval [0, 1] satisfying

$${\displaystyle \int _{0}^{1}{\bigl |}f(t){\bigr |}^{2}w(t)\,\mathrm {d} t<\infty }$$

is called the weighted L² space L²
_w([0, 1]), and w is called the weight function. The inner product is defined by

$${\displaystyle \langle f,g\rangle =\int _{0}^{1}f(t){\overline {g(t)}}w(t)\,\mathrm {d} t\,.}$$

The weighted space L²
_w([0, 1]) is identical with the Hilbert space L²([0, 1], μ) where the measure μ of a Lebesgue-measurable set A is defined by

$${\displaystyle \mu (A)=\int _{A}w(t)\,\mathrm {d} t\,.}$$

Weighted L² spaces like this are frequently used to study orthogonal polynomials, because different families of orthogonal polynomials are orthogonal with respect to different weighting functions.

Sobolev spaces, denoted by H^s or W^{s, 2}, are Hilbert spaces. These are a special kind of function space in which differentiation may be performed, but that (unlike other Banach spaces such as the Hölder spaces) support the structure of an inner product. Because differentiation is permitted, Sobolev spaces are a convenient setting for the theory of partial differential equations.[26] They also form the basis of the theory of direct methods in the calculus of variations.[27]

For s a non-negative integer and Ω ⊂ Rⁿ, the Sobolev space H^s(Ω) contains L² functions whose weak derivatives of order up to s are also L². The inner product in H^s(Ω) is

$${\displaystyle \langle f,g\rangle =\int _{\Omega }f(x){\bar {g}}(x)\,\mathrm {d} x+\int _{\Omega }Df(x)\cdot D{\bar {g}}(x)\,\mathrm {d} x+\cdots +\int _{\Omega }D^{s}f(x)\cdot D^{s}{\bar {g}}(x)\,\mathrm {d} x}$$

where the dot indicates the dot product in the Euclidean space of partial derivatives of each order. Sobolev spaces can also be defined when s is not an integer.

Sobolev spaces are also studied from the point of view of spectral theory, relying more specifically on the Hilbert space structure. If Ω is a suitable domain, then one can define the Sobolev space H^s(Ω) as the space of Bessel potentials;[28] roughly,

$${\displaystyle H^{s}(\Omega )=\left.\left\{(1-\Delta )^{-{\frac {s}{2}}}f\;\right|\,f\in L^{2}(\Omega )\right\}\,.}$$

Here Δ is the Laplacian and (1 − Δ)^−s/2 is understood in terms of the spectral mapping theorem. Apart from providing a workable definition of Sobolev spaces for non-integer s, this definition also has particularly desirable properties under the Fourier transform that make it ideal for the study of pseudodifferential operators. Using these methods on a compact Riemannian manifold, one can obtain for instance the Hodge decomposition, which is the basis of Hodge theory.[29]

The Hardy spaces are function spaces, arising in complex analysis and harmonic analysis, whose elements are certain holomorphic functions in a complex domain.[30] Let U denote the unit disc in the complex plane. Then the Hardy space H²(U) is defined as the space of holomorphic functions f on U such that the means

$${\displaystyle M_{r}(f)={\frac {1}{2\pi }}\int _{0}^{2\pi }\left|f\left(re^{i\theta }\right)\right|^{2}\,\mathrm {d} \theta }$$

remain bounded for r < 1. The norm on this Hardy space is defined by

$${\displaystyle \left\|f\right\|_{2}=\lim _{r\to 1}{\sqrt {M_{r}(f)}}\,.}$$

Hardy spaces in the disc are related to Fourier series. A function f is in H²(U) if and only if

$${\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}}$$

where

$${\displaystyle \sum _{n=0}^{\infty }|a_{n}|^{2}<\infty \,.}$$

Thus H²(U) consists of those functions that are L² on the circle, and whose negative frequency Fourier coefficients vanish.

The Bergman spaces are another family of Hilbert spaces of holomorphic functions.[31] Let D be a bounded open set in the complex plane (or a higher-dimensional complex space) and let L^{2, h}(D) be the space of holomorphic functions f in D that are also in L²(D) in the sense that

$${\displaystyle \|f\|^{2}=\int _{D}|f(z)|^{2}\,\mathrm {d} \mu (z)<\infty \,,}$$

where the integral is taken with respect to the Lebesgue measure in D. Clearly L^{2, h}(D) is a subspace of L²(D); in fact, it is a closed subspace, and so a Hilbert space in its own right. This is a consequence of the estimate, valid on compact subsets K of D, that

$${\displaystyle \sup _{z\in K}\left|f(z)\right|\leq C_{K}\left\|f\right\|_{2}\,,}$$

which in turn follows from Cauchy's integral formula. Thus convergence of a sequence of holomorphic functions in L²(D) implies also compact convergence, and so the limit function is also holomorphic. Another consequence of this inequality is that the linear functional that evaluates a function f at a point of D is actually continuous on L^{2, h}(D). The Riesz representation theorem implies that the evaluation functional can be represented as an element of L^{2, h}(D). Thus, for every z ∈ D, there is a function η_z ∈ L^{2, h}(D) such that

$${\displaystyle f(z)=\int _{D}f(\zeta ){\overline {\eta _{z}(\zeta )}}\,\mathrm {d} \mu (\zeta )}$$

for all f ∈ L^{2, h}(D). The integrand

$${\displaystyle K(\zeta ,z)={\overline {\eta _{z}(\zeta )}}}$$

is known as the Bergman kernel of D. This integral kernel satisfies a reproducing property

$${\displaystyle f(z)=\int _{D}f(\zeta )K(\zeta ,z)\,\mathrm {d} \mu (\zeta )\,.}$$

A Bergman space is an example of a reproducing kernel Hilbert space, which is a Hilbert space of functions along with a kernel K(ζ, z) that verifies a reproducing property analogous to this one. The Hardy space H²(D) also admits a reproducing kernel, known as the Szegő kernel.[32] Reproducing kernels are common in other areas of mathematics as well. For instance, in harmonic analysis the Poisson kernel is a reproducing kernel for the Hilbert space of square-integrable harmonic functions in the unit ball. That the latter is a Hilbert space at all is a consequence of the mean value theorem for harmonic functions.

The overtones of a vibrating string. These are eigenfunctions of an associated Sturm–Liouville problem. The eigenvalues 1, 1/2, 1/3, ... form the (musical) harmonic series.

In the theory of ordinary differential equations, spectral methods on a suitable Hilbert space are used to study the behavior of eigenvalues and eigenfunctions of differential equations. For example, the Sturm–Liouville problem arises in the study of the harmonics of waves in a violin string or a drum, and is a central problem in ordinary differential equations.[33] The problem is a differential equation of the form

$${\displaystyle -{\frac {\mathrm {d} }{\mathrm {d} x}}\left[p(x){\frac {\mathrm {d} y}{\mathrm {d} x}}\right]+q(x)y=\lambda w(x)y}$$

for an unknown function y on an interval [a, b], satisfying general homogeneous Robin boundary conditions

$${\displaystyle {\begin{cases}\alpha y(a)+\alpha 'y'(a)&=0\\\beta y(b)+\beta 'y'(b)&=0\,.\end{cases}}}$$

The functions p, q, and w are given in advance, and the problem is to find the function y and constants λ for which the equation has a solution. The problem only has solutions for certain values of λ, called eigenvalues of the system, and this is a consequence of the spectral theorem for compact operators applied to the integral operator defined by the Green's function for the system. Furthermore, another consequence of this general result is that the eigenvalues λ of the system can be arranged in an increasing sequence tending to infinity.[nb 2]

Hilbert spaces form a basic tool in the study of partial differential equations.[26] For many classes of partial differential equations, such as linear elliptic equations, it is possible to consider a generalized solution (known as a weak solution) by enlarging the class of functions. Many weak formulations involve the class of Sobolev functions, which is a Hilbert space. A suitable weak formulation reduces to a geometrical problem the analytic problem of finding a solution or, often what is more important, showing that a solution exists and is unique for given boundary data. For linear elliptic equations, one geometrical result that ensures unique solvability for a large class of problems is the Lax–Milgram theorem. This strategy forms the rudiment of the Galerkin method (a finite element method) for numerical solution of partial differential equations.[34]

A typical example is the Poisson equation −Δu = g with Dirichlet boundary conditions in a bounded domain Ω in R². The weak formulation consists of finding a function u such that, for all continuously differentiable functions v in Ω vanishing on the boundary:

$${\displaystyle \int _{\Omega }\nabla u\cdot \nabla v=\int _{\Omega }gv\,.}$$

This can be recast in terms of the Hilbert space H¹
₀(Ω) consisting of functions u such that u, along with its weak partial derivatives, are square integrable on Ω, and vanish on the boundary. The question then reduces to finding u in this space such that for all v in this space

$${\displaystyle a(u,v)=b(v)}$$

where a is a continuous bilinear form, and b is a continuous linear functional, given respectively by

$${\displaystyle a(u,v)=\int _{\Omega }\nabla u\cdot \nabla v,\quad b(v)=\int _{\Omega }gv\,.}$$

Since the Poisson equation is elliptic, it follows from Poincaré's inequality that the bilinear form a is coercive. The Lax–Milgram theorem then ensures the existence and uniqueness of solutions of this equation.

Hilbert spaces allow for many elliptic partial differential equations to be formulated in a similar way, and the Lax–Milgram theorem is then a basic tool in their analysis. With suitable modifications, similar techniques can be applied to parabolic partial differential equations and certain hyperbolic partial differential equations.

The path of a billiard ball in the Bunimovich stadium is described by an ergodic dynamical system.

The field of ergodic theory is the study of the long-term behavior of chaotic dynamical systems. The protypical case of a field that ergodic theory applies to is thermodynamics, in which—though the microscopic state of a system is extremely complicated (it is impossible to understand the ensemble of individual collisions between particles of matter)—the average behavior over sufficiently long time intervals is tractable. The laws of thermodynamics are assertions about such average behavior. In particular, one formulation of the zeroth law of thermodynamics asserts that over sufficiently long timescales, the only functionally independent measurement that one can make of a thermodynamic system in equilibrium is its total energy, in the form of temperature.

An ergodic dynamical system is one for which, apart from the energy—measured by the Hamiltonian—there are no other functionally independent conserved quantities on the phase space. More explicitly, suppose that the energy E is fixed, and let Ω_E be the subset of the phase space consisting of all states of energy E (an energy surface), and let T_t denote the evolution operator on the phase space. The dynamical system is ergodic if there are no continuous non-constant functions on Ω_E such that

$${\displaystyle f(T_{t}w)=f(w)}$$

for all w on Ω_E and all time t. Liouville's theorem implies that there exists a measure μ on the energy surface that is invariant under the time translation. As a result, time translation is a unitary transformation of the Hilbert space L²(Ω_E, μ) consisting of square-integrable functions on the energy surface Ω_E with respect to the inner product

$${\displaystyle \left\langle f,g\right\rangle _{L^{2}\left(\Omega _{E},\mu \right)}=\int _{E}f{\bar {g}}\,\mathrm {d} \mu \,.}$$

The von Neumann mean ergodic theorem[20] states the following:

• If U_t is a (strongly continuous) one-parameter semigroup of unitary operators on a Hilbert space H, and P is the orthogonal projection onto the space of common fixed points of U_t, {x ∈H | U_tx = x, ∀t > 0}, then
  $${\displaystyle Px=\lim _{T\to \infty }{\frac {1}{T}}\int _{0}^{T}U_{t}x\,\mathrm {d} t\,.}$$

  

For an ergodic system, the fixed set of the time evolution consists only of the constant functions, so the ergodic theorem implies the following:[35] for any function f ∈ L²(Ω_E, μ),

$${\displaystyle {\underset {T\to \infty }{L^{2}-\lim }}{\frac {1}{T}}\int _{0}^{T}f(T_{t}w)\,\mathrm {d} t=\int _{\Omega _{E}}f(y)\,\mathrm {d} \mu (y)\,.}$$

That is, the long time average of an observable f is equal to its expectation value over an energy surface.

Superposition of sinusoidal wave basis functions (bottom) to form a sawtooth wave (top)

Spherical harmonics, an orthonormal basis for the Hilbert space of square-integrable functions on the sphere, shown graphed along the radial direction

One of the basic goals of Fourier analysis is to decompose a function into a (possibly infinite) linear combination of given basis functions: the associated Fourier series. The classical Fourier series associated to a function f defined on the interval [0, 1] is a series of the form

$${\displaystyle \sum _{n=-\infty }^{\infty }a_{n}e^{2\pi in\theta }}$$

where

$${\displaystyle a_{n}=\int _{0}^{1}f(\theta )e^{-2\pi in\theta }\,\mathrm {d} \theta \,.}$$

The example of adding up the first few terms in a Fourier series for a sawtooth function is shown in the figure. The basis functions are sine waves with wavelengths λ/n (for integer n) shorter than the wavelength λ of the sawtooth itself (except for n = 1, the fundamental wave). All basis functions have nodes at the nodes of the sawtooth, but all but the fundamental have additional nodes. The oscillation of the summed terms about the sawtooth is called the Gibbs phenomenon.

A significant problem in classical Fourier series asks in what sense the Fourier series converges, if at all, to the function f. Hilbert space methods provide one possible answer to this question.[36] The functions e_n(θ) = e^2πinθ form an orthogonal basis of the Hilbert space L²([0, 1]). Consequently, any square-integrable function can be expressed as a series

$${\displaystyle f(\theta )=\sum _{n}a_{n}e_{n}(\theta )\,,\quad a_{n}=\langle f,e_{n}\rangle }$$

and, moreover, this series converges in the Hilbert space sense (that is, in the L² mean).

The problem can also be studied from the abstract point of view: every Hilbert space has an orthonormal basis, and every element of the Hilbert space can be written in a unique way as a sum of multiples of these basis elements. The coefficients appearing on these basis elements are sometimes known abstractly as the Fourier coefficients of the element of the space.[37] The abstraction is especially useful when it is more natural to use different basis functions for a space such as L²([0, 1]). In many circumstances, it is desirable not to decompose a function into trigonometric functions, but rather into orthogonal polynomials or wavelets for instance,[38] and in higher dimensions into spherical harmonics.[39]

For instance, if e_n are any orthonormal basis functions of L²[0, 1], then a given function in L²[0, 1] can be approximated as a finite linear combination[40]

$${\displaystyle f(x)\approx f_{n}(x)=a_{1}e_{1}(x)+a_{2}e_{2}(x)+\cdots +a_{n}e_{n}(x)\,.}$$

The coefficients {a_j} are selected to make the magnitude of the difference ||f − f_n||² as small as possible. Geometrically, the best approximation is the orthogonal projection of f onto the subspace consisting of all linear combinations of the {e_j}, and can be calculated by[41]

$${\displaystyle a_{j}=\int _{0}^{1}{\overline {e_{j}(x)}}f(x)\,\mathrm {d} x\,.}$$

That this formula minimizes the difference ||f − f_n||² is a consequence of Bessel's inequality and Parseval's formula.

In various applications to physical problems, a function can be decomposed into physically meaningful eigenfunctions of a differential operator (typically the Laplace operator): this forms the foundation for the spectral study of functions, in reference to the spectrum of the differential operator.[42] A concrete physical application involves the problem of hearing the shape of a drum: given the fundamental modes of vibration that a drumhead is capable of producing, can one infer the shape of the drum itself?[43] The mathematical formulation of this question involves the Dirichlet eigenvalues of the Laplace equation in the plane, that represent the fundamental modes of vibration in direct analogy with the integers that represent the fundamental modes of vibration of the violin string.

Spectral theory also underlies certain aspects of the Fourier transform of a function. Whereas Fourier analysis decomposes a function defined on a compact set into the discrete spectrum of the Laplacian (which corresponds to the vibrations of a violin string or drum), the Fourier transform of a function is the decomposition of a function defined on all of Euclidean space into its components in the continuous spectrum of the Laplacian. The Fourier transformation is also geometrical, in a sense made precise by the Plancherel theorem, that asserts that it is an isometry of one Hilbert space (the "time domain") with another (the "frequency domain"). This isometry property of the Fourier transformation is a recurring theme in abstract harmonic analysis (since it reflects the conservation of energy for the continuous Fourier Transform), as evidenced for instance by the Plancherel theorem for spherical functions occurring in noncommutative harmonic analysis.

The orbitals of an electron in a hydrogen atom are eigenfunctions of the energy.

In the mathematically rigorous formulation of quantum mechanics, developed by John von Neumann,[44] the possible states (more precisely, the pure states) of a quantum mechanical system are represented by unit vectors (called state vectors) residing in a complex separable Hilbert space, known as the state space, well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are points in the projectivization of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system; for example, the position and momentum states for a single non-relativistic spin zero particle is the space of all square-integrable functions, while the states for the spin of a single proton are unit elements of the two-dimensional complex Hilbert space of spinors. Each observable is represented by a self-adjoint linear operator acting on the state space. Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate.[45]

The inner product between two state vectors is a complex number known as a probability amplitude. During an ideal measurement of a quantum mechanical system, the probability that a system collapses from a given initial state to a particular eigenstate is given by the square of the absolute value of the probability amplitudes between the initial and final states.[46] The possible results of a measurement are the eigenvalues of the operator—which explains the choice of self-adjoint operators, for all the eigenvalues must be real. The probability distribution of an observable in a given state can be found by computing the spectral decomposition of the corresponding operator.[47]

For a general system, states are typically not pure, but instead are represented as statistical mixtures of pure states, or mixed states, given by density matrices: self-adjoint operators of trace one on a Hilbert space.[48] Moreover, for general quantum mechanical systems, the effects of a single measurement can influence other parts of a system in a manner that is described instead by a positive operator valued measure. Thus the structure both of the states and observables in the general theory is considerably more complicated than the idealization for pure states.[49]

Any true physical color can be represented by a combination of pure spectral colors. As physical colors can be composed of any number of spectral colors, the space of physical colors may aptly be represented by a Hilbert space over spectral colors. Humans have three types of cone cells for color perception, so the perceivable colors can be represented by 3-dimensional Euclidean space. The many-to-one linear mapping from the Hilbert space of physical colors to the Euclidean space of human perceivable colors explains why many distinct physical colors may be perceived by humans to be identical (e.g., pure yellow light versus a mix of red and green light, see metamerism).

Two vectors u and v in a Hilbert space H are orthogonal when ⟨u, v⟩ = 0. The notation for this is u ⊥ v. More generally, when S is a subset in H, the notation u ⊥ S means that u is orthogonal to every element from S.

When u and v are orthogonal, one has

$${\displaystyle \|u+v\|^{2}=\langle u+v,u+v\rangle =\langle u,u\rangle +2\,\operatorname {Re} \langle u,v\rangle +\langle v,v\rangle =\|u\|^{2}+\|v\|^{2}\,.}$$

By induction on n, this is extended to any family u₁, ..., u_n of n orthogonal vectors,

$${\displaystyle \left\|u_{1}+\cdots +u_{n}\right\|^{2}=\left\|u_{1}\right\|^{2}+\cdots +\left\|u_{n}\right\|^{2}.}$$

Whereas the Pythagorean identity as stated is valid in any inner product space, completeness is required for the extension of the Pythagorean identity to series. A series Σu_k of orthogonal vectors converges in H if and only if the series of squares of norms converges, and

$${\displaystyle \left\|\sum _{k=0}^{\infty }u_{k}\right\|^{2}=\sum _{k=0}^{\infty }\left\|u_{k}\right\|^{2}\,.}$$

Furthermore, the sum of a series of orthogonal vectors is independent of the order in which it is taken.

Geometrically, the parallelogram identity asserts that AC² + BD² = 2(AB² + AD²). In words, the sum of the squares of the diagonals is twice the sum of the squares of any two adjacent sides.

By definition, every Hilbert space is also a Banach space. Furthermore, in every Hilbert space the following parallelogram identity holds:

$${\displaystyle \left\|u+v\right\|^{2}+\left\|u-v\right\|^{2}=2\left(\left\|u\right\|^{2}+\left\|v\right\|^{2}\right)\,.}$$

Conversely, every Banach space in which the parallelogram identity holds is a Hilbert space, and the inner product is uniquely determined by the norm by the polarization identity.[50] For real Hilbert spaces, the polarization identity is

$${\displaystyle \langle u,v\rangle ={\frac {1}{4}}\left(\left\|u+v\right\|^{2}-\left\|u-v\right\|^{2}\right)\,.}$$

For complex Hilbert spaces, it is

$${\displaystyle \langle u,v\rangle ={\tfrac {1}{4}}\left(\left\|u+v\right\|^{2}-\left\|u-v\right\|^{2}+i\left\|u+iv\right\|^{2}-i\left\|u-iv\right\|^{2}\right)\,.}$$

The parallelogram law implies that any Hilbert space is a uniformly convex Banach space.[51]

This subsection employs the Hilbert projection theorem. If C is a non-empty closed convex subset of a Hilbert space H and x a point in H, there exists a unique point y ∈ C that minimizes the distance between x and points in C,[52]

$${\displaystyle y\in C\,,\quad \|x-y\|=\operatorname {dist} (x,C)=\min\{\|x-z\|:z\in C\}\,.}$$

This is equivalent to saying that there is a point with minimal norm in the translated convex set D = C − x. The proof consists in showing that every minimizing sequence (d_n) ⊂ D is Cauchy (using the parallelogram identity) hence converges (using completeness) to a point in D that has minimal norm. More generally, this holds in any uniformly convex Banach space.[53]

When this result is applied to a closed subspace F of H, it can be shown that the point y ∈ F closest to x is characterized by[54]

$${\displaystyle y\in F\,,\quad x-y\perp F\,.}$$

This point y is the orthogonal projection of x onto F, and the mapping P_F : x → y is linear (see Orthogonal complements and projections). This result is especially significant in applied mathematics, especially numerical analysis, where it forms the basis of least squares methods.[55]

In particular, when F is not equal to H, one can find a nonzero vector v orthogonal to F (select x ∉ F and v = x − y). A very useful criterion is obtained by applying this observation to the closed subspace F generated by a subset S of H.

A subset S of H spans a dense vector subspace if (and only if) the vector 0 is the sole vector v ∈ H orthogonal to S.

The dual space H* is the space of all continuous linear functions from the space H into the base field. It carries a natural norm, defined by

$${\displaystyle \|\varphi \|=\sup _{\|x\|=1,x\in H}|\varphi (x)|\,.}$$

This norm satisfies the parallelogram law, and so the dual space is also an inner product space where this inner product can be defined in terms of this dual norm by using the polarization identity. The dual space is also complete so it is a Hilbert space in its own right. If e_• = (e_i)_{i ∈ I} is a complete orthonormal basis for H then the inner product on the dual space of any two ${\displaystyle f,g\in H^{*}}$ is

$${\displaystyle \langle f,g\rangle _{H^{*}}=\sum _{i\in I}f(e_{i}){\overline {g(e_{i})}}}$$

where all but countably many of the terms in this series are zero.

The Riesz representation theorem affords a convenient description of the dual space. To every element u of H, there is a unique element φ_u of H*, defined by

$${\displaystyle \varphi _{u}(x)=\langle x,u\rangle }$$

where moreover, ${\displaystyle \left\|\varphi _{u}\right\|=\left\|u\right\|.}$

The Riesz representation theorem states that the map from H to H* defined by u ↦ φ_u is surjective, which makes this map an isometric antilinear isomorphism.[56] So to every element φ of the dual H* there exists one and only one u_φ in H such that

$${\displaystyle \langle x,u_{\varphi }\rangle =\varphi (x)}$$

for all x ∈ H. The inner product on the dual space H* satisfies

$${\displaystyle \langle \varphi ,\psi \rangle =\langle u_{\psi },u_{\varphi }\rangle \,.}$$

The reversal of order on the right-hand side restores linearity in φ from the antilinearity of u_φ. In the real case, the antilinear isomorphism from H to its dual is actually an isomorphism, and so real Hilbert spaces are naturally isomorphic to their own duals.

The representing vector u_φ is obtained in the following way. When φ ≠ 0, the kernel F = Ker(φ) is a closed vector subspace of H, not equal to H, hence there exists a nonzero vector v orthogonal to F. The vector u is a suitable scalar multiple λv of v. The requirement that φ(v) = ⟨v, u⟩ yields

$${\displaystyle u=\langle v,v\rangle ^{-1}\,{\overline {\varphi (v)}}\,v\,.}$$

This correspondence φ ↔ u is exploited by the bra–ket notation popular in physics. It is common in physics to assume that the inner product, denoted by ⟨x|y⟩, is linear on the right,

$${\displaystyle \langle x|y\rangle =\langle y,x\rangle \,.}$$

The result ⟨x|y⟩ can be seen as the action of the linear functional ⟨x| (the bra) on the vector |y⟩ (the ket).

The Riesz representation theorem relies fundamentally not just on the presence of an inner product, but also on the completeness of the space. In fact, the theorem implies that the topological dual of any inner product space can be identified with its completion. An immediate consequence of the Riesz representation theorem is also that a Hilbert space H is reflexive, meaning that the natural map from H into its double dual space is an isomorphism.

In a Hilbert space H, a sequence {x_n} is weakly convergent to a vector x ∈ H when

$${\displaystyle \lim _{n}\langle x_{n},v\rangle =\langle x,v\rangle }$$

for every v ∈ H.

For example, any orthonormal sequence {f_n} converges weakly to 0, as a consequence of Bessel's inequality. Every weakly convergent sequence {x_n} is bounded, by the uniform boundedness principle.

Conversely, every bounded sequence in a Hilbert space admits weakly convergent subsequences (Alaoglu's theorem).[57] This fact may be used to prove minimization results for continuous convex functionals, in the same way that the Bolzano–Weierstrass theorem is used for continuous functions on R^d. Among several variants, one simple statement is as follows:[58]

If f : H → R is a convex continuous function such that f(x) tends to +∞ when ||x|| tends to ∞, then f admits a minimum at some point x₀ ∈ H.

This fact (and its various generalizations) are fundamental for direct methods in the calculus of variations. Minimization results for convex functionals are also a direct consequence of the slightly more abstract fact that closed bounded convex subsets in a Hilbert space H are weakly compact, since H is reflexive. The existence of weakly convergent subsequences is a special case of the Eberlein–Šmulian theorem.

Any general property of Banach spaces continues to hold for Hilbert spaces. The open mapping theorem states that a continuous surjective linear transformation from one Banach space to another is an open mapping meaning that it sends open sets to open sets. A corollary is the bounded inverse theorem, that a continuous and bijective linear function from one Banach space to another is an isomorphism (that is, a continuous linear map whose inverse is also continuous). This theorem is considerably simpler to prove in the case of Hilbert spaces than in general Banach spaces.[59] The open mapping theorem is equivalent to the closed graph theorem, which asserts that a linear function from one Banach space to another is continuous if and only if its graph is a closed set.[60] In the case of Hilbert spaces, this is basic in the study of unbounded operators (see closed operator).

The (geometrical) Hahn–Banach theorem asserts that a closed convex set can be separated from any point outside it by means of a hyperplane of the Hilbert space. This is an immediate consequence of the best approximation property: if y is the element of a closed convex set F closest to x, then the separating hyperplane is the plane perpendicular to the segment xy passing through its midpoint.[61]

The continuous linear operators A : H₁ → H₂ from a Hilbert space H₁ to a second Hilbert space H₂ are bounded in the sense that they map bounded sets to bounded sets. Conversely, if an operator is bounded, then it is continuous. The space of such bounded linear operators has a norm, the operator norm given by

$${\displaystyle \lVert A\rVert =\sup \left\{\,\lVert Ax\rVert$$ :\lVert x\rVert \leq 1\,\right\}\,.}

The sum and the composite of two bounded linear operators is again bounded and linear. For y in H₂, the map that sends x ∈ H₁ to ⟨Ax, y⟩ is linear and continuous, and according to the Riesz representation theorem can therefore be represented in the form

$${\displaystyle \left\langle x,A^{*}y\right\rangle =\langle Ax,y\rangle }$$

for some vector A*y in H₁. This defines another bounded linear operator A* : H₂ → H₁, the adjoint of A. The adjoint satisfies A** = A. When the Riesz representation theorem is used to identify each Hilbert space with its continuous dual space, the adjoint of A can be shown to be identical to the transpose ^tA : H₂* → H₁* of A, which by definition sends ${\displaystyle \psi \in H_{2}^{*}}$ to the functional ${\displaystyle \psi \circ A\in H_{1}^{*}.}$

The set B(H) of all bounded linear operators on H (meaning operators H → H), together with the addition and composition operations, the norm and the adjoint operation, is a C*-algebra, which is a type of operator algebra.

An element A of B(H) is called 'self-adjoint' or 'Hermitian' if A* = A. If A is Hermitian and ⟨Ax, x⟩ ≥ 0 for every x, then A is called 'nonnegative', written A ≥ 0; if equality holds only when x = 0, then A is called 'positive'. The set of self adjoint operators admits a partial order, in which A ≥ B if A − B ≥ 0. If A has the form B*B for some B, then A is nonnegative; if B is invertible, then A is positive. A converse is also true in the sense that, for a non-negative operator A, there exists a unique non-negative square root B such that

$${\displaystyle A=B^{2}=B^{*}B\,.}$$

In a sense made precise by the spectral theorem, self-adjoint operators can usefully be thought of as operators that are "real". An element A of B(H) is called normal if A*A = AA*. Normal operators decompose into the sum of a self-adjoint operator and an imaginary multiple of a self adjoint operator

$${\displaystyle A={\frac {A+A^{*}}{2}}+i{\frac {A-A^{*}}{2i}}}$$

that commute with each other. Normal operators can also usefully be thought of in terms of their real and imaginary parts.

An element U of B(H) is called unitary if U is invertible and its inverse is given by U*. This can also be expressed by requiring that U be onto and ⟨Ux, Uy⟩ = ⟨x, y⟩ for all x, y ∈ H. The unitary operators form a group under composition, which is the isometry group of H.

An element of B(H) is compact if it sends bounded sets to relatively compact sets. Equivalently, a bounded operator T is compact if, for any bounded sequence {x_k}, the sequence {Tx_k} has a convergent subsequence. Many integral operators are compact, and in fact define a special class of operators known as Hilbert–Schmidt operators that are especially important in the study of integral equations. Fredholm operators differ from a compact operator by a multiple of the identity, and are equivalently characterized as operators with a finite dimensional kernel and cokernel. The index of a Fredholm operator T is defined by

$${\displaystyle \operatorname {index} T=\dim \ker T-\dim \operatorname {coker} T\,.}$$

The index is homotopy invariant, and plays a deep role in differential geometry via the Atiyah–Singer index theorem.

Unbounded operators are also tractable in Hilbert spaces, and have important applications to quantum mechanics.[62] An unbounded operator T on a Hilbert space H is defined as a linear operator whose domain D(T) is a linear subspace of H. Often the domain D(T) is a dense subspace of H, in which case T is known as a densely defined operator.

The adjoint of a densely defined unbounded operator is defined in essentially the same manner as for bounded operators. Self-adjoint unbounded operators play the role of the observables in the mathematical formulation of quantum mechanics. Examples of self-adjoint unbounded operators on the Hilbert space L²(R) are:[63]

• A suitable extension of the differential operator
  $${\displaystyle (Af)(x)=-i{\frac {\mathrm {d} }{\mathrm {d} x}}f(x)\,,}$$

  
  where i is the imaginary unit and f is a differentiable function of compact support.
• The multiplication-by-x operator:
  $${\displaystyle (Bf)(x)=xf(x)\,.}$$

  

These correspond to the momentum and position observables, respectively. Note that neither A nor B is defined on all of H, since in the case of A the derivative need not exist, and in the case of B the product function need not be square integrable. In both cases, the set of possible arguments form dense subspaces of L²(R).

Two Hilbert spaces H₁ and H₂ can be combined into another Hilbert space, called the (orthogonal) direct sum,[64] and denoted

$${\displaystyle H_{1}\oplus H_{2}\,,}$$

consisting of the set of all ordered pairs (x₁, x₂) where x_i ∈ H_i, i = 1, 2, and inner product defined by

$${\displaystyle {\bigl \langle }(x_{1},x_{2}),(y_{1},y_{2}){\bigr \rangle }_{H_{1}\oplus H_{2}}=\left\langle x_{1},y_{1}\right\rangle _{H_{1}}+\left\langle x_{2},y_{2}\right\rangle _{H_{2}}\,.}$$

More generally, if H_i is a family of Hilbert spaces indexed by i ∈ I, then the direct sum of the H_i, denoted

$${\displaystyle \bigoplus _{i\in I}H_{i}}$$

consists of the set of all indexed families

$${\displaystyle x=(x_{i}\in H_{i}|i\in I)\in \prod _{i\in I}H_{i}}$$

in the Cartesian product of the H_i such that

$${\displaystyle \sum _{i\in I}\|x_{i}\|^{2}<\infty \,.}$$

The inner product is defined by

$${\displaystyle \langle x,y\rangle =\sum _{i\in I}\left\langle x_{i},y_{i}\right\rangle _{H_{i}}\,.}$$

Each of the H_i is included as a closed subspace in the direct sum of all of the H_i. Moreover, the H_i are pairwise orthogonal. Conversely, if there is a system of closed subspaces, V_i, i ∈ I, in a Hilbert space H, that are pairwise orthogonal and whose union is dense in H, then H is canonically isomorphic to the direct sum of V_i. In this case, H is called the internal direct sum of the V_i. A direct sum (internal or external) is also equipped with a family of orthogonal projections E_i onto the ith direct summand H_i. These projections are bounded, self-adjoint, idempotent operators that satisfy the orthogonality condition

$${\displaystyle E_{i}E_{j}=0,\quad i\neq j\,.}$$

The spectral theorem for compact self-adjoint operators on a Hilbert space H states that H splits into an orthogonal direct sum of the eigenspaces of an operator, and also gives an explicit decomposition of the operator as a sum of projections onto the eigenspaces. The direct sum of Hilbert spaces also appears in quantum mechanics as the Fock space of a system containing a variable number of particles, where each Hilbert space in the direct sum corresponds to an additional degree of freedom for the quantum mechanical system. In representation theory, the Peter–Weyl theorem guarantees that any unitary representation of a compact group on a Hilbert space splits as the direct sum of finite-dimensional representations.

If x₁, y₁ ∊ H₁ and x₂, y₂ ∊ H₂, then one defines an inner product on the (ordinary) tensor product as follows. On simple tensors, let

$${\displaystyle \langle x_{1}\otimes x_{2},\,y_{1}\otimes y_{2}\rangle =\langle x_{1},y_{1}\rangle \,\langle x_{2},y_{2}\rangle \,.}$$

This formula then extends by sesquilinearity to an inner product on H₁ ⊗ H₂. The Hilbertian tensor product of H₁ and H₂, sometimes denoted by H₁ ${\displaystyle {\widehat {\otimes }}}$ H₂, is the Hilbert space obtained by completing H₁ ⊗ H₂ for the metric associated to this inner product.[65]

An example is provided by the Hilbert space L²([0, 1]). The Hilbertian tensor product of two copies of L²([0, 1]) is isometrically and linearly isomorphic to the space L²([0, 1]²) of square-integrable functions on the square [0, 1]². This isomorphism sends a simple tensor f₁ ⊗ f₂ to the function

$${\displaystyle (s,t)\mapsto f_{1}(s)\,f_{2}(t)}$$

on the square.

This example is typical in the following sense.[66] Associated to every simple tensor product x₁ ⊗ x₂ is the rank one operator from H^∗
₁ to H₂ that maps a given x* ∈ H^∗
₁ as

$${\displaystyle x^{*}\mapsto x^{*}(x_{1})x_{2}\,.}$$

This mapping defined on simple tensors extends to a linear identification between H₁ ⊗ H₂ and the space of finite rank operators from H^∗
₁ to H₂. This extends to a linear isometry of the Hilbertian tensor product H₁ ${\displaystyle {\widehat {\otimes }}}$ H₂ with the Hilbert space HS(H^∗
₁, H₂) of Hilbert–Schmidt operators from H^∗
₁ to H₂.

The space ${\displaystyle \ell _{2}}$ of square-summable sequences of complex numbers is the set of infinite sequences

$${\displaystyle (c_{1},c_{2},c_{3},\dots )}$$

of real or complex numbers such that

$${\displaystyle \left|c_{1}\right|^{2}+\left|c_{2}\right|^{2}+\left|c_{3}\right|^{2}+\cdots <\infty \,.}$$

This space has an orthonormal basis:

$${\displaystyle {\begin{aligned}e_{1}&=(1,0,0,\dots )\\e_{2}&=(0,1,0,\dots )\\&\ \ \vdots \end{aligned}}}$$

This space is the infinite-dimensional generalization of the ${\displaystyle \ell _{2}^{n}}$ space of finite-dimensional vectors. It is usually the first example used to show that in infinite-dimensional spaces, a set that is closed and bounded is not necessarily (sequentially) compact (as is the case in all finite dimensional spaces). Indeed, the set of orthonormal vectors above shows this: It is an infinite sequence of vectors in the unit ball (i.e., the ball of points with norm less than or equal one). This set is clearly bounded and closed; yet, no subsequence of these vectors converges to anything and consequently the unit ball in ${\displaystyle \ell _{2}}$ is not compact. Intuitively, this is because "there is always another coordinate direction" into which the next elements of the sequence can evade.

One can generalize the space ${\displaystyle \ell _{2}}$ in many ways. For example, if B is any (infinite) set, then one can form a Hilbert space of sequences with index set B, defined by

$${\displaystyle \ell ^{2}(B)=\left\{x:B{\xrightarrow {x}}\mathbb {C} \,\left|\,\sum _{b\in B}\left|x(b)\right|^{2}<\infty \right.\right\}\,.}$$

The summation over B is here defined by

$${\displaystyle \sum _{b\in B}\left|x(b)\right|^{2}=\sup \sum _{n=1}^{N}\left|x(b_{n})\right|^{2}}$$

the supremum being taken over all finite subsets of B. It follows that, for this sum to be finite, every element of l²(B) has only countably many nonzero terms. This space becomes a Hilbert space with the inner product

$${\displaystyle \langle x,y\rangle =\sum _{b\in B}x(b){\overline {y(b)}}}$$

for all x, y ∈ l²(B). Here the sum also has only countably many nonzero terms, and is unconditionally convergent by the Cauchy–Schwarz inequality.

An orthonormal basis of l²(B) is indexed by the set B, given by

$${\displaystyle e_{b}(b')={\begin{cases}1&{\text{if }}b=b'\\0&{\text{otherwise.}}\end{cases}}}$$

Let f₁, …, f_n be a finite orthonormal system in H. For an arbitrary vector x ∈ H, let

$${\displaystyle y=\sum _{j=1}^{n}\langle x,f_{j}\rangle \,f_{j}\,.}$$

Then ⟨x, f_k⟩ = ⟨y, f_k⟩ for every k = 1, …, n. It follows that x − y is orthogonal to each f_k, hence x − y is orthogonal to y. Using the Pythagorean identity twice, it follows that

$${\displaystyle \|x\|^{2}=\|x-y\|^{2}+\|y\|^{2}\geq \|y\|^{2}=\sum _{j=1}^{n}{\bigl |}\langle x,f_{j}\rangle {\bigr |}^{2}\,.}$$

Let {f_i}, i ∈ I, be an arbitrary orthonormal system in H. Applying the preceding inequality to every finite subset J of I gives Bessel's inequality:[68]

$${\displaystyle \sum _{i\in I}{\bigl |}\langle x,f_{i}\rangle {\bigr |}^{2}\leq \|x\|^{2},\quad x\in H}$$

(according to the definition of the sum of an arbitrary family of non-negative real numbers).

Geometrically, Bessel's inequality implies that the orthogonal projection of x onto the linear subspace spanned by the f_i has norm that does not exceed that of x. In two dimensions, this is the assertion that the length of the leg of a right triangle may not exceed the length of the hypotenuse.

Bessel's inequality is a stepping stone to the stronger result called Parseval's identity, which governs the case when Bessel's inequality is actually an equality. By definition, if {e_k}_{k ∈ B} is an orthonormal basis of H, then every element x of H may be written as

$${\displaystyle x=\sum _{k\in B}\left\langle x,e_{k}\right\rangle \,e_{k}\,.}$$

Even if B is uncountable, Bessel's inequality guarantees that the expression is well-defined and consists only of countably many nonzero terms. This sum is called the Fourier expansion of x, and the individual coefficients ⟨x, e_k⟩ are the Fourier coefficients of x. Parseval's identity then asserts that

$${\displaystyle \|x\|^{2}=\sum _{k\in B}|\langle x,e_{k}\rangle |^{2}\,.}$$

Conversely, if {e_k} is an orthonormal set such that Parseval's identity holds for every x, then {e_k} is an orthonormal basis.

As a consequence of Zorn's lemma, every Hilbert space admits an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same cardinality, called the Hilbert dimension of the space.[69] For instance, since l²(B) has an orthonormal basis indexed by B, its Hilbert dimension is the cardinality of B (which may be a finite integer, or a countable or uncountable cardinal number).

As a consequence of Parseval's identity, if {e_k}_{k ∈ B} is an orthonormal basis of H, then the map Φ : H → l²(B) defined by Φ(x) = ⟨x, e_k⟩_k∈B is an isometric isomorphism of Hilbert spaces: it is a bijective linear mapping such that

$${\displaystyle {\bigl \langle }\Phi (x),\Phi (y){\bigr \rangle }_{l^{2}(B)}=\left\langle x,y\right\rangle _{H}}$$

for all x, y ∈ H. The cardinal number of B is the Hilbert dimension of H. Thus every Hilbert space is isometrically isomorphic to a sequence space l²(B) for some set B.

By definition, a Hilbert space is separable provided it contains a dense countable subset. Along with Zorn's lemma, this means a Hilbert space is separable if and only if it admits a countable orthonormal basis. All infinite-dimensional separable Hilbert spaces are therefore isometrically isomorphic to l².

In the past, Hilbert spaces were often required to be separable as part of the definition.[70] Most spaces used in physics are separable, and since these are all isomorphic to each other, one often refers to any infinite-dimensional separable Hilbert space as "the Hilbert space" or just "Hilbert space".[71] Even in quantum field theory, most of the Hilbert spaces are in fact separable, as stipulated by the Wightman axioms. However, it is sometimes argued that non-separable Hilbert spaces are also important in quantum field theory, roughly because the systems in the theory possess an infinite number of degrees of freedom and any infinite Hilbert tensor product (of spaces of dimension greater than one) is non-separable.[72] For instance, a bosonic field can be naturally thought of as an element of a tensor product whose factors represent harmonic oscillators at each point of space. From this perspective, the natural state space of a boson might seem to be a non-separable space.[72] However, it is only a small separable subspace of the full tensor product that can contain physically meaningful fields (on which the observables can be defined). Another non-separable Hilbert space models the state of an infinite collection of particles in an unbounded region of space. An orthonormal basis of the space is indexed by the density of the particles, a continuous parameter, and since the set of possible densities is uncountable, the basis is not countable.[72]