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.

Mimicking the definition for matrices, a bounded linear operator A over a separable Hilbert space H is said to be in the trace class if for some ( and hence all ) orthonormal bases

With suitable restrictions, much can be said about the structure of the spectra of transformations in a Hilbert space.

David Hilbert is said to have remarked that the theory of complex multiplication of elliptic curves was not only the most beautiful part of mathematics but of all science.

In operator theory, a bounded operator T on a Hilbert space is said to be nilpotent if T < sup > n </ sup > = 0 for some n. It is said to be quasinilpotent or topological nilpotent if its spectrum σ ( T ) =

Jason Spencer of the Houston Chronicle said that the current principal and formal principal said that Bellaire has " reputation for academic excellence " because, in the words of former principal Hilbert Bludau, " Parents felt ownership of that school.

A sequence of points in a Hilbert space H is said to converge weakly to a point x in H if

Nearly simultaneously David Hilbert published " The Foundations of Physics ", an axiomatic derivation of the field equations ( see Einstein – Hilbert action ).

When his colleague Richard Courant wrote the now classic Methods of Mathematical Physics including some of Hilbert's ideas, he added Hilbert's name as author even though Hilbert had not directly contributed to the writing.

* R. Courant, D. Hilbert, Methods of Mathematical Physics, vol II.

* Methods of Mathematical Physics by R. Courant, D. Hilbert ISBN 0-471-50447-5 ( Volume 1 Paperback ) ISBN 0-471-50439-4 ( Volume 2 Paperback ) ISBN 0-471-17990-6 ( Hardback )

He and David Hilbert authored the influential textbook Methods of Mathematical Physics, which is still widely used more than eighty years after it was written.

The converse is not always true ; not every Banach space is a Hilbert space.

A necessary and sufficient condition for a Banach space X to be associated to an inner product ( which will then necessarily make X into a Hilbert space ) is the parallelogram identity:

So, for example, while R < sup > n </ sup > is a Banach space with respect to any norm defined on it, it is only a Hilbert space with respect to the Euclidean norm.

Similarly, as an infinite-dimensional example, the Lebesgue space L < sup > p </ sup > is always a Banach space but is only a Hilbert space when p = 2.

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.

Bra-ket notation can be used even if the vector space is not a Hilbert space.

* Dynamics is also described by linear operators on the Hilbert space.

The Hilbert space of a spin-0 point particle is spanned by a " position basis ", where the label r extends over the set of all points in space.

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

A different basis for the same Hilbert space is:

In an N-dimensional Hilbert space, can be written as an N × 1 column vector, and then A is an N × N matrix with complex entries.

In an N-dimensional Hilbert space, can be written as a 1 × N row vector, and A ( as in the previous section ) is an N × N matrix.

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.

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

Let be a Hilbert space and is a vector in.

* C *- algebra: A Banach algebra that is a closed *- subalgebra of the algebra of bounded operators on some Hilbert space.

By a theorem of Gelfand and Naimark, given a B * algebra A there exists a Hilbert space H and an isometric *- homomorphism from A into the algebra B ( H ) of all bounded linear operators on H. Thus every B * algebra is isometrically *- isomorphic to a C *- algebra.

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,

Later this was criticised, for example by Vladimir Arnold, as too much Hilbert, not enough Poincaré.

Varying over the entire Hilbert space is usually too complicated for physical calculations, and a subspace of the entire Hilbert space is chosen, parametrized by some ( real ) differentiable parameters α < sub > i </ sub > ( i = 1, 2 .., N ).

The original statement of Hilbert, however, has also been judged too vague to admit a definitive answer.

If H is a Hilbert space with inner product, then so too is the kth exterior power with inner product

( It is hard to tell exactly what Hilbert was saying, one problem being that he may have been using the term " elliptic function " to mean both the elliptic function ℘ and the elliptic modular function j.

Impagliazzo's contributions to the field of computational complexity include: the construction of a pseudorandom number generator from any one-way function, his proof of the XOR lemma via " hard core sets ", his work on break through results in propositional proof complexity, such as the exponential size lower bound for constant-depth Hilbert proofs of the pigeonhole principle and the introduction of the polynomial calculus system, his work on connections between computational hardness and derandomization, and a recent break-through work on the construction of multi-source seedless extractors.

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

Consider a complete orthonormal system ( basis ),, for a Hilbert space H, with respect to the norm from an inner product.

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 ).

A non-expansive mapping with can be strengthened to a firmly non-expansive mapping in a Hilbert space H if the following holds for all x and y in H:

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.

Hilbert is known as one of the founders of proof theory and mathematical logic, as well as for being among the first to distinguish between mathematics and metamathematics.

In the spring of 1882, Hermann Minkowski ( two years younger than Hilbert and also a native of Königsberg but so talented he had graduated early from his gymnasium and gone to Berlin for three semesters ), returned to Königsberg and entered the university.

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.

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.

Starting with Moritz Pasch in 1882, many improved axiomatic systems for geometry have been proposed, the best known being those of Hilbert, George Birkhoff, and Tarski.

In mathematics and computer science, the (, German for ' decision problem ') is a challenge posed by David Hilbert in 1928.

The spin degree of freedom for an electron is associated with a two-dimensional complex Hilbert space H, with each quantum state corresponding to a vector in that space.

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