30 Facts About Hilbert space

The success of Hilbert space methods ushered in a very fruitful era for functional analysis.

An element of a Hilbert space can be uniquely specified by its coordinates with respect to an orthonormal basis, in analogy with Cartesian coordinates in classical geometry.

One of the most familiar examples of a Hilbert space is the Euclidean vector space consisting of three-dimensional vectors, denoted by, and equipped with the dot product.

Multivariable calculus in Euclidean Hilbert space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist.

Hilbert space is a real or complex inner product space that is a complete metric space with respect to the distance function induced by the inner product.

Related searches

Real inner product Hilbert space is defined in the same way, except that is a real vector Hilbert space and the inner product takes real values.

Completeness of is expressed using a form of the Cauchy criterion for sequences in : a pre-Hilbert space is complete if every Cauchy sequence converges with respect to this norm to an element in the space.

The proof is basic in mathematical analysis, and permits mathematical series of elements of the Hilbert space to be manipulated with the same ease as series of complex numbers .

In particular, the idea of an abstract linear Hilbert space had gained some traction towards the end of the 19th century: this is a Hilbert space whose elements can be added together and multiplied by scalars without necessarily identifying these elements with "geometric" vectors, such as position and momentum vectors in physical systems.

In 1907, Frigyes Riesz and Ernst Sigismund Fischer independently proved that the Hilbert space of square Lebesgue-integrable functions is a complete metric Hilbert space.

The name "Hilbert space" was adopted by others, for example by Hermann Weyl in his book on quantum mechanics and the theory of groups.

Significance of the concept of a Hilbert space was underlined with the realization that it offers one of the best mathematical formulations of quantum mechanics.

Algebra of observables in quantum mechanics is naturally an algebra of operators defined on a Hilbert space, according to Werner Heisenberg's matrix mechanics formulation of quantum theory.

Bergman space is an example of a reproducing kernel Hilbert space, which is a Hilbert space of functions along with a kernel that verifies a reproducing property analogous to this one.

The Hardy Hilbert space admits a reproducing kernel, known as the Szego kernel.

That the latter is a Hilbert space at all is a consequence of the mean value theorem for harmonic functions.

Hilbert space methods provide one possible answer to this question.

Problem can be studied from the abstract point of view: 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 basis elements.

The exact nature of this Hilbert space is dependent on the system; for example, the position and momentum states for a single non-relativistic spin zero particle is the space of all square-integrable functions, while the states for the spin of a single proton are unit elements of the two-dimensional complex Hilbert space of spinors.

Whereas the Pythagorean identity as stated is valid in any inner product Hilbert space, completeness is required for the extension of the Pythagorean identity to series.

Parallelogram law implies that any Hilbert space is a uniformly convex Banach space.

An immediate consequence of the Riesz representation theorem is that a Hilbert space is reflexive, meaning that the natural map from into its double dual space is an isomorphism.

The open mapping theorem states that a continuous surjective linear transformation from one Banach Hilbert space to another is an open mapping meaning that it sends open sets to open sets.

The open mapping theorem is equivalent to the closed graph theorem, which asserts that a linear function from one Banach Hilbert space to another is continuous if and only if its graph is a closed set.

The Hilbertian tensor product of and, sometimes denoted by H2, is the Hilbert space obtained by completing for the metric associated to this inner product.

Related searches

That the span of the basis vectors is dense implies that every vector in the Hilbert space can be written as the sum of an infinite series, and the orthogonality implies that this decomposition is unique.

An orthonormal basis of the Hilbert space is indexed by the density of the particles, a continuous parameter, and since the set of possible densities is uncountable, the basis is not countable.

Every closed subspace V of a Hilbert space is therefore the image of an operator of norm one such that.

The closure of a subHilbert space can be completely characterized in terms of the orthogonal complement: if is a subHilbert space of, then the closure of is equal to.

Hilbert space-Spaess is first described as "a ubiquitous double agent" and later as "at least a double agent".