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evidence: The mathematical concept of a Hilbert space , named after David Hilbert , generalizes the notion of Euclidean space .. mathematical. mathematics. David Hilbert. David Hilbert. Euclidean space. Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions .. vector algebra. linear algebra. calculus. calculus. Euclidean plane. plane ( geometry ). A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured .. vector space. vector space. structure. mathematical structure. inner product. inner product space. Furthermore , Hilbert spaces are complete  : there are enough limits in the space to allow the techniques of calculus to be used .. calculus. calculus. complete. Complete metric space. limits. limit ( mathematics ). Hilbert spaces arise naturally and frequently in mathematics and physics , typically as infinite-dimensional function spaces .. mathematics. mathematics. physics. physics. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert , Erhard Schmidt , and Frigyes Riesz .. David Hilbert. David Hilbert. Erhard Schmidt. Erhard Schmidt. Frigyes Riesz. Frigyes Riesz. They are indispensable tools in the theories of partial differential equations , quantum mechanics , Fourier analysis -LRB- which includes applications to signal processing and heat transfer -RRB- -- and ergodic theory , which forms the mathematical underpinning of thermodynamics .. mathematical. mathematics. quantum mechanics. mathematical formulation of quantum mechanics. Fourier analysis. Fourier analysis. signal processing. signal processing. ergodic theory. ergodic theory. thermodynamics. thermodynamics. John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications .. John von Neumann. John von Neumann. The success of Hilbert space methods ushered in a very fruitful era for functional analysis .. functional analysis. functional analysis. Apart from the classical Euclidean spaces , examples of Hilbert spaces include spaces of square-integrable functions , spaces of sequences , Sobolev spaces consisting of generalized functions , and Hardy spaces of holomorphic functions .. spaces of square-integrable functions. Lp space. spaces of sequences. Sequence space. Geometric intuition plays an important role in many aspects of Hilbert space theory .. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space .. Pythagorean theorem. Pythagorean theorem. parallelogram law. parallelogram law. At a deeper level , perpendicular projection onto a subspace -LRB- the analog of `` dropping the altitude '' of a triangle -RRB- plays a significant role in optimization problems and other aspects of the theory .. dropping the altitude. altitude ( triangle ). An element of a Hilbert space can be uniquely specified by its coordinates with respect to a set of coordinate axes -LRB- an orthonormal basis -RRB- , in analogy with Cartesian coordinates in the plane .. coordinate axes. coordinate axes. orthonormal basis. orthonormal basis. When that set of axes is countably infinite , this means that the Hilbert space can also usefully be thought of in terms of the space of infinite sequences that are square-summable .. countably infinite. countably infinite. square-summable. Lp norm. The latter space is often in the older literature referred to as the Hilbert space .. Linear operators on a Hilbert space are likewise fairly concrete objects  : in good cases , they are simply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectrum .. spectrum. spectral theory