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Hilbert SpaceEdit

Hilbert space is a central construct in mathematics that generalizes the familiar geometry of finite-dimensional Euclidean spaces to infinite dimensions. At its core is an inner product, a rule that assigns a length to each vector and a measure of the angle between pairs of vectors. When every Cauchy sequence with respect to the induced norm converges within the space, we say the space is complete. This combination of an inner product and completeness gives Hilbert spaces a geometric feel while remaining entirely abstract, which is why they appear in a wide range of settings from pure analysis to physics and engineering.

In practice, Hilbert spaces provide a rigorous setting for questions about approximation, orthogonality, and spectral behavior of operators. They underpin the mathematical formulation of quantum mechanics, support techniques in signal processing, and are a natural habitat for many results in functional analysis and numerical analysis. For many problems, moving from finite-dimensional intuition to the infinite-dimensional realm is made possible by the structure of a Hilbert space.

Definition

A Hilbert space H is a vector space over the real or complex numbers equipped with an inner product ⟨•,•⟩ that induces a norm by ‖x‖ = sqrt(⟨x,x⟩). The inner product is linear in one argument, conjugate symmetric, and positive definite. The defining property of a Hilbert space is completeness with respect to the metric induced by this norm: every Cauchy sequence in H converges to a limit in H. When H is over the complex numbers one often writes ⟨x,y⟩, while for a real Hilbert space the same symbol is used with the usual real inner product.

For a vector x in H and a subspace M ⊆ H, the notions of length, angle, projection, and orthogonality generalize in the same way as in ℝ^n, with the inner product providing the quantitative content of these ideas. The concepts of orthogonality and projection are especially important for approximation and decomposition problems.

Examples

  • Finite-dimensional Euclidean spaces: R^n with the standard inner product ⟨x,y⟩ = ∑ x_i y_i are Hilbert spaces. This gives the concrete geometric intuition most readers start with.

  • The space of square-summable sequences, denoted by l2 space, consists of all sequences (x_n) with ∑ |x_n|^2 < ∞, equipped with ⟨x,y⟩ = ∑ x_n · conjugate(y_n). It is a classic example of a separable Hilbert space.

  • The space of square-integrable functions on a measure space, L2 space, consists of all measurable functions f with ∫ |f|^2 dμ < ∞, with inner product ⟨f,g⟩ = ∫ f · conjugate(g) dμ. Common special cases include L2([0,1]) and L2(ℝ) with the Lebesgue measure.

  • Subspaces of these spaces, closed under the same inner product, are themselves Hilbert spaces. In particular, closed subspaces of a Hilbert space are complete with respect to the restricted inner product.

Key properties

  • Orthogonality and projections: Given a closed subspace M ⊆ H, for every x ∈ H there exists a unique y ∈ M such that x − y is orthogonal to M. The operator P_M: H → M mapping x to this y is called the orthogonal projection onto M.

  • Cauchy-Schwarz inequality and orthonormality: For all x,y ∈ H, |⟨x,y⟩| ≤ ‖x‖‖y‖, with equality precisely when x and y are linearly dependent. An orthonormal set {e_k} in H satisfies ⟨e_j,e_k⟩ = δ_jk.

  • Orthonormal bases and expansions: In a separable Hilbert space, there exists an orthonormal basis {e_k} such that every x ∈ H can be written as a (potentially infinite) sum x = ∑ ⟨x,e_k⟩ e_k with convergence in norm. Parseval’s identity relates the norm of x to the sum of squared coefficients: ‖x‖^2 = ∑ |⟨x,e_k⟩|^2.

  • Riesz representation theorem: Every continuous linear functional f on H can be represented as f(x) = ⟨x, h_f⟩ for a unique h_f ∈ H. This ties the dual space closely to H itself and is a cornerstone of functional analysis.

Orthonormal bases and decomposition

In many Hilbert spaces of interest, one can choose a countable orthonormal basis. The Gram–Schmidt process provides a method to produce such a basis from any linearly independent set. Once an orthonormal basis is in place, many problems reduce to analyzing the coefficients ⟨x,e_k⟩ and exploiting the completeness to ensure convergence of expansions.

Operators on Hilbert space

  • Bounded linear operators: A linear map T: H → H is bounded if there exists a constant C with ‖Tx‖ ≤ C‖x‖ for all x ∈ H. Boundedness guarantees continuity.

  • Adjoint: Every bounded linear operator T has a unique adjoint T* characterized by ⟨Tx,y⟩ = ⟨x,T*y⟩ for all x,y ∈ H. This generalizes transpose and complex conjugate operations.

  • Special classes:

    • Self-adjoint (T = T*): Generalizing symmetric matrices; spectra lie on the real axis for such operators.
    • Unitary (T*T = TT* = I): Preserve norms and inner products, analogous to rotations.
    • Normal (T*T = TT*): Includes self-adjoint and unitary operators as important subclasses.

  • Spectrum and spectral theorem: The spectrum of an operator captures its generalized eigenvalues. The spectral theorem provides a powerful representation for self-adjoint and unitary operators, enabling functional calculus: for suitable functions f, one can define f(T) in a meaningful way.

  • Compact operators: Operators that map the unit ball to a relatively compact set. They generalize finite-rank operators and have discrete spectra with possible accumulation only at zero.

Applications

  • Quantum mechanics: The state space of a quantum system is modeled by a Hilbert space, with observables represented by self-adjoint operators and measurements described via spectral properties.

  • Signal processing and Fourier analysis: L2-based techniques underpin Fourier series and transforms; projection onto subspaces corresponds to filtering and approximation.

  • Machine learning and statistics: Kernel methods and reproducing kernel Hilbert spaces provide a framework for learning and function approximation in high or infinite dimensions.

  • Probability: Random processes with finite second moments naturally take values in a real or complex Hilbert space; the inner product corresponds to covariance, and completeness supports limit theorems in a functional-analytic setting.

See also