Hilbert space

A complete vector space equipped with an inner product, which defines lengths and angles and induces a norm. Completeness ensures that every Cauchy sequence converges within the space.

Inner product space: A vector space equipped with an inner product that defines lengths and angles. A Hilbert space is an inner product space that also satisfies completeness.
Orthogonal projection: The nearest-point map onto a closed subspace of an inner product space. Completeness guarantees that every closed subspace has a well-defined nearest-point projection.
Quantum mechanics: A physical theory in which states and observables are represented using mathematical structures including Hilbert spaces. Quantum state vectors occupy Hilbert spaces, with inner products encoding probability amplitudes.
Lp space: A function space defined by integrability of the p-th power of a function's magnitude. For p other than 2, the usual Lp norm generally does not arise from an inner product.
Normed vector space: A vector space equipped with a norm that measures vector length. Every inner product induces a norm, but not every norm comes from an inner product.
Orthonormal basis: A complete orthonormal family whose vectors span a Hilbert space in the limit. Coordinates along an orthonormal basis represent vectors and preserve their inner products.
Least squares: A method for fitting a model by minimizing the sum of squared residuals. In Hilbert spaces, least-squares solutions are projections onto subspaces of candidate models.
Pre-Hilbert space: An inner product space that is not necessarily complete in its induced norm. A pre-Hilbert space lacks the completeness condition that defines a Hilbert space.
Cauchy sequence: A sequence whose terms become arbitrarily close to one another as the sequence progresses. Completeness requires every Cauchy sequence to converge to a vector in the space.
Riesz representation theorem: A theorem identifying continuous linear functionals on a Hilbert space with inner products against fixed vectors. It turns abstract linear functionals into concrete vectors using the space's inner product.

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