Some Hilbert spaces, like, have finite dimension, while others have infinite dimension.

However, the definition of " Hilbert space " can be broadened to accommodate these states ( see the Gelfand – Naimark – Segal construction or rigged Hilbert spaces ).

Banach spaces are a different generalization of Hilbert spaces.

Two Hilbert spaces V and W may form a third space by a tensor product.

If a system is composed of two subsystems described in V and W respectively, then the Hilbert space of the entire system is the tensor product of the two spaces.

When applied to vector spaces of functions ( which typically are infinite-dimensional ), dual spaces are employed for defining and studying concepts like measures, distributions, and Hilbert spaces.

He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis.

The Hilbert space of the electron pair is, the tensor product of the two electrons ' Hilbert spaces.

An important object of study in functional analysis are the continuous linear operators defined on Banach and Hilbert spaces.

Hilbert spaces can be completely classified: there is a unique Hilbert space up to isomorphism for every cardinality of the orthonormal basis.

Finite-dimensional Hilbert spaces are fully understood in linear algebra, and infinite-dimensional separable Hilbert spaces are isomorphic to.

Separability being important for applications, functional analysis of Hilbert spaces consequently mostly deals with this space.

General Banach spaces are more complicated than Hilbert spaces, and cannot be classified in such a simple manner as those.

Fourier series can be conveniently studied in the context of Hilbert spaces, which provides a connection between harmonic analysis and functional analysis.

Every Hilbert space X is a Banach space because, by definition, a Hilbert space is complete with respect to the norm associated with its inner product, where a norm and an inner product are said to be associated if for all x ∈ X.

( Technically, the quantum states are rays of vectors in the Hilbert space, as corresponds to the same state for any nonzero complex number c .)

* Measurements are associated with linear operators ( called observables ) on the Hilbert space of quantum states.

Since there are infinitely many vectors in the basis, this is an infinite-dimensional Hilbert space.

The object physicists are considering when using the " bra-ket " notation is a Hilbert space ( a complete inner product space ).

are just different notations for expressing an inner product between two elements in a Hilbert space ( or for the first three, in any inner product space ).

Although Kronecker had conceded, Hilbert would later respond to others ' similar criticisms that " many different constructions are subsumed under one fundamental idea " — in other words ( to quote Reid ): " Through a proof of existence, Hilbert had been able to obtain a construction "; " the proof " ( i. e. the symbols on the page ) was " the object ".

Hilbert did not work in the central areas of analytic number theory, but his name has become known for the Hilbert – Pólya conjecture, for reasons that are anecdotal.

In the infinite-dimensional case, additional structures are important ( e. g. whether or not the space is a Hilbert space, Banach space, etc.

In contrast all finite-dimensional inner product spaces over or, such as those used in quantum computation, are automatically metrically complete and hence Hilbert spaces.

In other words, symmetric and antisymmetric states are essentially unchanged under the exchange of particle labels: they are only multiplied by a factor of + 1 or − 1, rather than being " rotated " somewhere else in the Hilbert space.

On the Infinite was Hilbert ’ s most important paper on the foundations of mathematics, serving as the heart of Hilbert's program to secure the foundation of transfinite numbers by basing them on finite methods.

These conditions played an important role in the development of the structure theory of commutative rings in the works of David Hilbert, Emmy Noether, and Emil Artin.

Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics.

At the University of Göttingen, Hilbert was surrounded by a social circle of some of the most important mathematicians of the 20th century, such as Emmy Noether and Alonzo Church.

One who had to leave Germany, Paul Bernays, had collaborated with Hilbert in mathematical logic, and co-authored with him the important book Grundlagen der Mathematik ( which eventually appeared in two volumes, in 1934 and 1939 ).

Around 1909, Hilbert dedicated himself to the study of differential and integral equations ; his work had direct consequences for important parts of modern functional analysis.

His work was a key aspect of Hermann Weyl and John von Neumann's work on the mathematical equivalence of Werner Heisenberg's matrix mechanics and Erwin Schrödinger's wave equation and his namesake Hilbert space plays an important part in quantum theory.

An important example is a Hilbert space, where the norm arises from an inner product.

This theorem establishes an important connection between a Hilbert space and its ( continuous ) dual space: if the underlying field is the real numbers, the two are isometrically isomorphic ; if the field is the complex numbers, the two are isometrically anti-isomorphic.

Generalisations of this identity play an important role in the theory of the Hilbert transform.

His work is also seen as important as a first step towards the theory of Hilbert spaces.

This relationship was discovered by David Hilbert who proved Nullstellensatz and several other important related theorems named after him ( like Hilbert's basis theorem ).

This is somewhat ironic, since arguably Weil was the mathematician of the 1940s and 1950s who best played the Hilbert role, being conversant with nearly all areas of ( theoretical ) mathematics and having been important in the development of many of them.

It is however much more general as there are important infinite-dimensional Hilbert spaces.

The convergence of this infinite sum is important if is to be a Hilbert space.

It is an important theoretical tool in the theory of integral equations ; it is used in the Hilbert space theory of stochastic processes, for example the Karhunen-Loève theorem ; and it is also used to characterize a symmetric positive semi-definite kernel.

The ring comes with the Hilbert polynomial P, an important invariant ( depending on embedding ) of X.

The Hilbert transform is also important in the field of signal processing where it is used to derive the analytic representation of a signal u ( t ).

An important subset of the reproducing kernel Hilbert spaces are the reproducing kernel Hilbert spaces associated to a continuous kernel.

Together with David Hilbert he made important contributions to functional analysis.