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.

Although Schrödinger himself after a year proved the equivalence of his wave-mechanics and Heisenberg's matrix mechanics, the reconciliation of the two approaches and their modern abstraction as motions in Hilbert space is generally attributed to Paul Dirac, who wrote a lucid account in his 1930 classic Principles of Quantum Mechanics.

Several of the Hilbert problems have been resolved ( or arguably resolved ) in ways that would have been profoundly surprising, and even disturbing, to Hilbert himself.

David Hilbert himself devoted much of his research to the sixth problem ; in particular, he worked in those fields of physics that arose after he stated the problem.

Hilbert himself proved the finite generation of invariant rings in the case of the field of complex numbers for some classical semi-simple Lie groups ( in particular the general linear group over the complex numbers ) and specific linear actions on polynomial rings, i. e. actions coming from finite-dimensional representations of the Lie-group.

Bassermann refused to take the ring back and, at the second meeting, declared that Hilbert could do with the ring as he pleased.

If the norm of a Banach space satisfies this identity, the associated inner product which makes it into a Hilbert space is given by the polarization identity.

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.

If the earlier completion procedure is applied to a normed vector space, the result is a Banach space containing the original space as a dense subspace, and if it is applied to an inner product space, the result is a Hilbert space containing the original space as a dense subspace.

If the underlying manifold is allowed to be infinite dimensional ( for example, a Hilbert manifold ), then one arrives at the notion of an infinite-dimensional Lie group.

If ρ acts on a finite dimensional Hilbert space and has eigenvalues

Let H be a Hilbert space, and let H * denote its dual space, consisting of all continuous linear functionals from H into the field R or C. If x is an element of H, then the function φ < sub > x </ sub >, defined by

In 1900, David Hilbert posed an influential question about transcendental numbers, Hilbert's seventh problem: If a is an algebraic number, that is not zero or one, and b is an irrational algebraic number, is a < sup > b </ sup > necessarily transcendental?

If R is a ring, let R denote the ring of polynomials in the indeterminate X over R. Hilbert proved that if R is " not too large ", in the sense that if R is Noetherian, the same must be true for R. Formally,

* If R is a Noetherian ring, then R is Noetherian by the Hilbert basis theorem.

If the kernel used is a Gaussian radial basis function, the corresponding feature space is a Hilbert space of infinite dimensions.

If a normal operator on a finite-dimensional real or complex Hilbert space ( inner product space ) stabilizes a subspace, then it also stabilizes its orthogonal complement.

If X is a Hilbert space and T is a normal operator, then a remarkable result known as the spectral theorem gives an analogue of the diagonalisation theorem for normal finite-dimensional operators ( Hermitian matrices, for example ).

If a point in the Hilbert cube is specified by a sequence with, then a homeomorphism to the infinite dimensional unit cube is given by.

If A has a multiplicative identity 1, then it is immediate that the equivalence class ξ in the GNS Hilbert space H containing 1 is a cyclic vector for the above representation.

If X is the toric variety corresponding to the normal fan of P, then P defines an ample line bundle on X, and the Ehrhart polynomial of P coincides with the Hilbert polynomial of this line bundle.

If one wants to consider antipodal points as identified, one passes to projective space ( see also projective Hilbert space, for this idea as applied in quantum mechanics ).

If, then the distinguished vector in the Hilbert space is thought of as the vacuum state defined by.

If is a bounded linear operator on a Hilbert space, then this notion coincides with the condition that

If is a quadratic irrational, then the j-invariant is an algebraic integer of degree, the class number of and the minimal ( monic integral ) polynomial it satisfies is called the Hilbert class polynomial.

If one thinks of operators on a Hilbert space as " generalized complex numbers ", then the adjoint of an operator plays the role of the complex conjugate of a complex number.

Tentative Proof: If the underlying Hilbert space is finite-dimensional, the spectral theorem says that N is of the form

If it does not converge, is no longer well-defined, but a modified definition where one integrates over arbitrarily large, relatively compact domains, still yields the Einstein equation as the Euler – Lagrange equation of the Einstein – Hilbert action.

If A < sub > 1 </ sub >, ..., A < sub > n </ sub > are n operators each localized in a bounded region and U ( a ) represents the unitary operator actively translating the Hilbert space by the vector a, then if we pick some subset of the n operators to translate,

If U is an isometric map defined on a closed subset H < sub > 1 </ sub > of a Hilbert space H then we can define an extension W of U to all of H by the condition that W be zero on the orthogonal complement of H < sub > 1 </ sub >.

The term-algebra was introduced by I. E. Segal in 1947 to describe norm-closed subalgebras of, namely, the space of bounded operators on some Hilbert space.

* Together with the identity matrix I ( which is sometimes written as σ < sub > 0 </ sub >), the Pauli matrices form an orthogonal basis, in the sense of Hilbert-Schmidt, for the real Hilbert space of 2 × 2 complex Hermitian matrices, or the complex Hilbert space of all 2 × 2 matrices.

, the Richmond City Council consisted of: Kathy C. Graziano, 4th District, President of Council ; Ellen F. Robertson, 6th District, Vice-President of Council ; Bruce Tyler, 1st District ; Charles R. Samuels, 2nd District ; Chris A. Hilbert, 3rd District ; E. Martin ( Marty ) Jewell, 5th District ; Cynthia I Newbille, 7th District ; Reva M. Trammell, 8th District ; and Douglas G. Conner Jr., 9th District.

Thus we define the representations of G on an Hilbert space H to be those group homomorphisms, ρ, which arise from continuous actions of G on H. We say that a representation ρ is unitary if ρ ( g ) is a unitary operator for all g ∈ G ; i. e., for all v, w ∈ H. ( I. e.

The Cauchy completion of A / I in the quotient norm is a Hilbert space H.

The work of David Hilbert, proving that I ( V ) was finitely presented in many cases, almost put an end to classical invariant theory for several decades, though the classical epoch in the subject continued to the final publications of Alfred Young, more than 50 years later.

For a number of years, he had been an assistant to Richard Courant at Göttingen in the preparation of Courant and David Hilbert ’ s book Methoden der mathematischen Physik I, which was published in 1924.

* N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Pitman, 1981.

The unit cube I < sup > n </ sup > as well as the Hilbert cube I < sup > ω </ sup > are absolute retracts.

Let H be a Hilbert space and G the set of bounded invertible operators on H of the form I + T, where T is a trace-class operator.

In the late 1920s, the mathematicians Gabriel Sudan and Wilhelm Ackermann, students of David Hilbert, were studying the foundations of computation.

Among the students of Hilbert were: Hermann Weyl, chess champion Emanuel Lasker, Ernst Zermelo, and Carl Gustav Hempel.

Hilbert and the mathematicians who worked with him in his enterprise were committed to the project.

In 1926 von Neuman showed that if atomic states were understood as vectors in Hilbert space, then they would correspond with both Schrödinger's wave function theory and Heisenberg's matrices.

The concepts were highly influential, and his own contribution lives on in the names of the Hilbert class field and of the Hilbert symbol of local class field theory.

One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N. Whitehead and David Hilbert, were pioneering the use of logic and set theory to understand the foundations of mathematics.

In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues ; more precisely: as spectral values ( point spectrum plus absolute continuous plus singular continuous spectrum ) of linear operators in Hilbert space.

In his above-mentioned account, he introduced the bra-ket notation, together with an abstract formulation in terms of the Hilbert space used in functional analysis ; he showed that Schrödinger's and Heisenberg's approaches were two different representations of the same theory, and found a third, most general one, which represented the dynamics of the system.

These views were forcefully expressed by David Hilbert in 1928, when he wrote in < cite > Die Grundlagen der Mathematik </ cite >, " Taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope to the astronomer or to the boxer the use of his fists ".

The foundations of quantum mechanics were established during the first half of the 20th century by Max Planck, Niels Bohr, Werner Heisenberg, Louis de Broglie, Arthur Compton, Albert Einstein, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Enrico Fermi, Wolfgang Pauli, David Hilbert, Wilhelm Wien, Satyendra Nath Bose, Arnold Sommerfeld and others.

As with the Hilbert problems, one of the prize problems ( the Poincaré conjecture ) was solved relatively soon after the problems were announced.

And on the side of modular forms, there were examples such as Hilbert modular forms, Siegel modular forms, and theta-series.

The historical development of commutative algebra, which was initially called ideal theory, is closely linked to concepts in elimination theory: ideas of Kronecker, who wrote a major paper on the subject, were adapted by Hilbert and effectively ' linearised ' while dropping the explicit constructive content.

Now, intuitively groups were conceived as rotations on geometric objects, so it is only natural to study representations which essentially arise from continuous actions on Hilbert spaces.

( For those who were first introduced to dual groups consisting of characters which are the continuous homomorphisms into the circle group, this approach is similar except that the circle group is ( ultimately ) generalised to the group of unitary operators on a given Hilbert space.

These conditions were first identified by David Hilbert and Paul Bernays in their Grundlagen der Arithmetik.

Hilbert had investigated the M-curves of degree 6, and found that the 11 components always were grouped in a certain way.

In 1926, John von Neumann became assistant to David Hilbert, and he would coin the term Hilbert space to describe the algebra and analysis which were used in the development of quantum mechanics.

Academic contacts in Germany: Carathéodory's contacts in Germany were many and included such famous names as: Minkowski, Hilbert, Klein, Einstein, Schwarz, Fejér.

Bateman, perhaps influenced by Hilbert ’ s point of view in mathematical physics as a whole, was the first to see that the basic ideas of electromagnetism were equivalent to statements regarding integrals of differential forms, statements for which Grassmann's calculus of extension on differentiable manifolds, Poincaré's theories of Stokesian transformations and integral invariants, and Lie's theory of continuous groups could be fruitfully applied.

Hopf spent the year after his doctorate at Göttingen, where David Hilbert, Richard Courant, Carl Runge, and Emmy Noether were working.

In mathematics, Hilbert's program, formulated by German mathematician David Hilbert, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies.