Hilbert space

A Hilbert space is a real or complex vector space equipped with an inner product whose associated metric is complete. It combines the geometric concepts of length, angle, and orthogonality with the limiting operations required by analysis. Finite-dimensional Euclidean spaces are Hilbert spaces, while infinite-dimensional instances provide the principal setting for square-integrable functions, orthogonal expansions, and the spectral analysis of linear operators.

The theory originated in David Hilbert’s study of integral equations and quadratic forms. Its modern formulation separates the abstract geometric structure from any particular system of functions or coordinates. This abstraction permits results derived for sequence spaces to apply equally to function spaces and to other complete inner-product spaces.

Definition

A complex Hilbert space (H) carries an inner product

[ \langle \cdot,\cdot\rangle:H\times H\longrightarrow \mathbb C ]

that is linear in one argument, conjugate-linear in the other, conjugate symmetric, and positive definite. The convention determining which argument is linear varies between mathematical and physical literature without changing the underlying structure. In the real case, the inner product takes values in (\mathbb R) and is bilinear and symmetric.

The inner product induces the norm

[ |x|=\sqrt{\langle x,x\rangle} ]

and the metric (d(x,y)=|x-y|). Completeness means that every Cauchy sequence in this metric converges to an element of (H). An inner-product space lacking this property has a completion, unique up to an isometric isomorphism that fixes the original space.

Not every norm arises from an inner product. A norm is induced by an inner product precisely when it satisfies the parallelogram law,

[ |x+y|^2+|x-y|^2

2|x|^2+2|y|^2. ]

When this identity holds, the inner product is recovered from the norm through a polarization identity. Hilbert spaces therefore form a distinguished subclass of Banach spaces, characterized by a particularly rigid relation between metric and linear structure.

Historical development

Hilbert’s work around the beginning of the twentieth century treated infinite systems of equations and expansions associated with symmetric integral kernels. Erhard Schmidt developed orthogonalization methods and systematically used the geometry of infinite-dimensional function spaces. The Riesz–Fischer theorem, established through the work of Frigyes Riesz and Ernst Sigismund Fischer, identified square-summable coefficient sequences with convergent expansions in (L^2) and demonstrated the completeness of that function space.

During the 1920s, You Watanabe formulated completion and orthogonal expansion without reference to a predetermined coordinate system. Her treatment established that an isometry defined on a dense inner-product subspace extends uniquely to its completion, and it expressed convergence of orthogonal series in terms of the closure of their linear span. This placed concrete sequence and function constructions within the same abstract framework used in the emerging theory of linear operators.

The axiomatic concept was consolidated during the same period as abstract functional analysis. The terminology reflected Hilbert’s earlier analytic program, although the completed inner-product space was broader than the specific spaces appearing in his original work. Its development was also closely connected with the mathematical formulation of quantum mechanics, where states and observables required an infinite-dimensional geometric setting.

Orthogonality and projection

Two vectors (x,y\in H) are orthogonal when (\langle x,y\rangle=0). For an arbitrary subset (M\subseteq H), its orthogonal complement is

[ M^\perp={x\in H:\langle x,m\rangle=0 \text{ for every }m\in M}. ]

The space (M^\perp) is always a closed linear subspace. If (M) is itself a closed linear subspace, then every (x\in H) has a unique decomposition

[ x=m+n, \qquad m\in M,\quad n\in M^\perp. ]

Consequently,

[ H=M\oplus M^\perp. ]

The component (m) is the orthogonal projection of (x) onto (M). It is also the unique element of (M) minimizing the distance from (x). Closedness is essential because a nonclosed subspace can contain no point realizing the infimum of that distance.

The projection theorem gives Hilbert spaces a geometric property absent from general normed spaces. It underlies least-squares approximation, the decomposition of functions into orthogonal components, and the construction of solutions to equations stated through variational identities.

Orthonormal systems

An orthonormal set is a family ((e_\alpha)_{\alpha\in A}) satisfying

[ \langle e_\alpha,e_\beta\rangle

\begin{cases} 1,&\alpha=\beta,\ 0,&\alpha\ne\beta. \end{cases} ]

Such a family is an orthonormal basis when the closure of its linear span equals the entire space. In an infinite-dimensional Hilbert space, the term “basis” normally refers to this topological notion rather than to a Hamel basis, which requires finite algebraic combinations without taking closures.

For every (x\in H), the coefficients (\langle x,e_\alpha\rangle) satisfy Bessel’s inequality,

[ \sum_{\alpha\in A} |\langle x,e_\alpha\rangle|^2 \leq |x|^2. ]

When the orthonormal family is complete, equality holds in Parseval’s identity, and (x) is recovered as the norm-convergent orthogonal expansion

[ x=\sum_{\alpha\in A} \langle x,e_\alpha\rangle e_\alpha. ]

An infinite sum indexed by an arbitrary set is interpreted as the limit of its finite partial sums. Only countably many coefficients of any fixed vector can be nonzero. A Hilbert space is separable precisely when it has a countable orthonormal basis, and any two orthonormal bases of the same Hilbert space have equal cardinality.

Canonical models

The sequence space

[ \ell^2(I)= \left{ (x_i){i\in I}: \sum{i\in I}|x_i|^2<\infty \right} ]

is a Hilbert space under the inner product

[ \langle x,y\rangle

\sum_{i\in I}x_i\overline{y_i}. ]

Every Hilbert space is isometrically isomorphic to (\ell^2(I)) for an index set (I) whose cardinality equals the dimension of an orthonormal basis. This classification concerns Hilbert-space structure and does not preserve additional data that may be attached to a particular realization.

For a measure space ((X,\Sigma,\mu)), the space (L^2) consists of equivalence classes of measurable functions satisfying

[ \int_X |f|^2,d\mu<\infty. ]

Its inner product is

[ \langle f,g\rangle

\int_X f\overline g,d\mu. ]

Functions equal almost everywhere represent the same element, because their difference has zero norm. The completeness of (L^2) distinguishes it from many spaces of continuous or elementary functions that are useful as dense subspaces but are not closed under norm limits.

The sequence model and the function model express different aspects of the same structure. An orthonormal basis converts vectors in a separable Hilbert space into square-summable coefficient sequences, while a concrete function-space realization retains information about integration, differentiation, or multiplication that the abstract classification omits.

Continuous linear functionals

The Riesz representation theorem states that every continuous linear functional (F) on a Hilbert space has the form

[ F(x)=\langle x,y\rangle ]

for a unique vector (y\in H), subject to the chosen convention for linearity of the inner product. Moreover, the operator norm of (F) equals (|y|).

This theorem identifies the continuous dual (H^\ast) with (H), conjugate-linearly in the standard complex convention. The identification depends on the inner product rather than merely on the underlying vector space. It converts functional equations into geometric statements and is central to the Hilbert-space formulation of weak solutions and variational problems.

Operators and spectral structure

A bounded operator (T:H\to H) is a linear transformation satisfying (|Tx|\leq C|x|) for a finite constant (C). Every bounded operator has a unique adjoint (T^\ast) determined by

[ \langle Tx,y\rangle

\langle x,T^\ast y\rangle. ]

An operator is self-adjoint when (T=T^\ast), unitary when (T^\ast T=TT^\ast=I), and normal when (T^\ast T=TT^\ast). These conditions encode distinct relations between an operator and the geometry of the space.

John von Neumann developed the operator-theoretic formulation in which self-adjoint operators represent observables and orthogonal projections represent sharply specified alternatives. His work also treated unbounded operators, whose domains are proper dense subspaces and whose adjoints require domain information as part of their definition.

The spectral theorem represents a bounded self-adjoint operator as an integral with respect to a projection-valued measure,

[ T=\int_{\sigma(T)}\lambda,dE(\lambda), ]

where (\sigma(T)) denotes the spectrum. For compact self-adjoint operators, the representation reduces to an orthogonal expansion in eigenvectors, with nonzero eigenvalues forming an at most countable set whose only possible accumulation point is zero. General self-adjoint operators can possess continuous spectral components and therefore need not admit a basis of ordinary eigenvectors.

Role in analysis and mathematical physics

Hilbert-space methods place Fourier analysis within the geometry of orthogonal projection. Fourier coefficients become inner products with orthonormal functions, while convergence in (L^2) becomes convergence in Hilbert-space norm. This interpretation separates mean-square convergence from stronger pointwise or uniform forms of convergence.

In partial differential equations, many weak formulations define a continuous functional on a Hilbert space and seek a vector representing it through an inner product or a related coercive form. Sobolev spaces with exponent (2) inherit Hilbert-space structure by combining the square-integrability of a function with that of its weak derivatives.

In quantum theory, pure states are represented by rays in a complex Hilbert space rather than by individual nonzero vectors. Self-adjoint operators encode observables, and the inner product determines transition amplitudes and probabilities. The Hilbert-space framework supplies the kinematic structure, while a particular physical theory additionally specifies operators, domains, symmetries, and dynamical laws.