Hilbert space

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The mathematical concept of a Hilbert space generalizes the notion of Euclidean space. It extends the methods of Euclidean geometry and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces of any finite or infinite dimension. A Hilbert space is an abstract vector space, and it has the additional structure of an inner product that allows length and angle to be measured. Finally, Hilbert spaces are required to be complete, a property that stipulates the existence of enough limits in the space to allow the techniques of calculus to be used. Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert (after whom they are named), Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics...

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