Hilbert space

For the Hilbert space-filling curve, see Hilbert curve.

Hilbert spaces can be used to study the harmonics of vibrating strings.

A Hilbert space is a mathematical concept covering the extra-dimensional use of Euclidean space—i.e., a space with more than three dimensions. A Hilbert space uses the mathematics of two and three dimensions to try and describe what happens in greater than three dimensions. It is named after David Hilbert.

Vector algebra and calculus are methods normally used in the two-dimensional Euclidean plane and three-dimensional space. In Hilbert spaces, these methods can be used with any finite or infinite number of dimensions. A Hilbert space is a vector space that has the structure of an inner product that allows length and angle to be measured. Hilbert spaces also have to be complete, which means that enough limits have to exist for calculus to work.

The earliest Hilbert spaces were studied in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. John von Neumann first came up with the name "Hilbert Space". Hilbert space methods made a big difference to functional analysis.

Hilbert spaces show up a lot in mathematics, physics, and engineering, often as infinite-dimensional function spaces. They are especially useful for studying partial differential equations, quantum mechanics, Fourier analysis (which includes signal processing and heat transfer). Hilbert spaces are used in ergodic theory which is the mathematical basis of thermodynamics. All normal Euclidean spaces are also Hilbert spaces. Other examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces made up of generalized functions, and Hardy spaces of holomorphic functions.

Hilbert Space Media

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  The state of a vibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinct overtones is given by the projection of the point onto the coordinate axes in the space.

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  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).

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  Triangle inequality in a metric space

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  David Hilbert, wearing a dashing hat

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

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

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

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

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  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.

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Other websites

• Hilbert space at Mathworld
• 245B, notes 5: Hilbert spaces by Terence Tao