Hilbert space

A Hilbert space is a vector space equipped with an inner product whose induced metric is complete.¹ The inner product assigns a scalar to two vectors and determines their norm by ||x|| = √⟨x,x⟩; it also defines orthogonality, angles, and distances. Completeness means that every Cauchy sequence of vectors converges to a vector that remains in the space.

Finite-dimensional Euclidean spaces are Hilbert spaces, but the concept becomes especially useful in infinite dimensions. Standard examples include the sequence space ℓ², consisting of square-summable sequences, and the space L² of square-integrable functions. The parallelogram law characterizes norms that arise from an inner product, distinguishing Hilbert geometry from more general Banach-space geometry.

Hilbert-space geometry turns approximation into a precise theory of orthogonal projection. Every closed subspace has a unique nearest-point projection for each vector, a fact underlying least-squares methods, Fourier analysis, and the decomposition of signals into orthogonal components.

An orthonormal basis allows vectors to be represented by coordinates, including infinite coordinate lists, and Parseval's identity relates the squared norm of a vector to the sum of squared coefficients. Bounded linear operators on a Hilbert space have adjoints, defined through the inner product; self-adjoint, unitary, normal, and compact operators form especially important classes. The spectral theorem extends diagonalization from matrices to suitable operators, sometimes using a continuous spectrum rather than ordinary eigenvectors.

In the standard mathematical formulation of quantum mechanics, a physical pure state is represented by a ray in a complex Hilbert space, while observables are represented by self-adjoint operators and measurement outcomes by their spectra.² Vectors that differ only by a nonzero complex scalar describe the same pure state, so the physical state space is a projective space rather than the Hilbert space itself.

The inner product supplies transition amplitudes and probabilities through the Born rule. A wavefunction is commonly an element of L²(R³), although position eigenstates used in calculations are generalized vectors rather than ordinary square-integrable vectors. Composite systems are modeled with tensor products of Hilbert spaces, producing entanglement; time evolution of an isolated system is represented by a unitary operator, equivalently by the Schrödinger equation under suitable conditions.⁴

Hilbert spaces also support subtle distinctions that are easy to miss in elementary treatments. A separable Hilbert space has a countable dense subset and therefore, in the infinite-dimensional case, a countable orthonormal basis; nonseparable Hilbert spaces exist and matter in some functional-analytic constructions. Completeness is not cosmetic: completing a pre-Hilbert space can add limits of physically or mathematically meaningful approximations.

Not every linear operator appearing in quantum theory is bounded. Position and momentum are typically unbounded and require carefully specified domains, a source of technical issues involving self-adjoint extensions and boundary conditions. Gleason's theorem shows, under dimensional and regularity assumptions, how probability measures on subspaces are constrained by the Hilbert-space structure.³ Hilbert spaces also appear in reproducing-kernel methods, harmonic analysis, control theory, and numerical approximation, where orthogonal projection supplies stable representations.

Inner product

A sesquilinear or bilinear pairing that defines length, angle, and orthogonality.

Complete

Having the property that every Cauchy sequence converges to an element of the space.

Orthonormal basis

A complete orthonormal family used to expand vectors into coordinates.

Self-adjoint operator

An operator equal to its adjoint, with domain conditions included when it is unbounded.

Separable

Containing a countable dense subset; equivalently, having a countable orthonormal basis for a Hilbert space.