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Every inner product <.,.> on a real or complex vector space H gives rise to a norm ||.|| as follows:
We call H a Hilbert space if it is complete with respect to this norm. Completeness in this context means that any Cauchy sequence of elements of the space converges to an element in the space, in the sense that the norm of differences approaches zero. Every Hilbert space is thus also a Banach space (but not vice versa).
All finite-dimensional inner product spaces (such as Euclidean spaceEuclidean space is the usual n dimensional mathematical space, a generalization of the 2- and 3-dimensional spaces studied by Euclid. Formally, for any non-negative integer n n dimensional Euclidean space is the set R n (where R is the set of real numbers with the ordinary dot productIn mathematics, the dot product (also known as the scalar product and the inner product is a function (·) : V × V → F, where V is a vector space and F its underlying field. In other words, it maps a pair of vectors to a scalar. When the latter term i) are Hilbert spaces. However, the infinite-dimensional examples are much more important in applications. These applications include:
The inner product allows one to adopt a "geometrical" view and use geometrical language familiar from finite dimensional spaces. Of all the infinite-dimensional topological vector spaces, the Hilbert spaces are the most " well-behaved" and the closest to the finite-dimensional spaces.
The elements of an abstract Hilbert space are sometimes called "vectors". In applications, they are typically sequences of complex numbers or functions. In quantum mechanics for example, a physical system is described by a complex Hilbert space which contains the " wavefunctions" that stand for the possible states of the system. See mathematical formulation of quantum mechanics.
One goal of Fourier analysis is to write a given function as a (possibly infinite) sum of multiples of given base functions. This problem can be studied abstractly in Hilbert spaces: every Hilbert space has an orthonormal basis, and every element of the Hilbert space can be written in a unique way as a sum of multiples of these base elements.
Hilbert spaces were named after David Hilbert, who studied them in the context of integral equations. The origin of the designation, however is unclear, but it was already used by Hermann Weyl in his famous book The Theory of Groups and Quantum Mechanics published in 1931. John von Neumann was perhaps the mathematician who most clearly recognized their importance.
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