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 every 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 th ...

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Hilbert space, Hilbert space - Introduction, Hilbert space - Definition, Hilbert space - Examples, Hilbert space - Euclidean spaces, Hilbert space - Sequence spaces, Hilbert space - Lebesgue spaces, Hilbert space - Sobolev spaces, Hilbert space - Operations on Hilbert spaces, Hilbert space - Bases, Hilbert space - Orthogonal complements and projections, Hilbert space - Reflexivity, Hilbert space - Bounded operators, Hilbert space - Unbounded operators

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Hilbert space, Hilbert space - Introduction, Hilbert space - Definition, Hilbert space - Examples, Hilbert space - Euclidean spaces, Hilbert space - Sequence spaces, Hilbert space - Lebesgue spaces, Hilbert space - Sobolev spaces, Hilbert space - Operations on Hilbert spaces, Hilbert space - Bases, Hilbert space - Orthogonal complements and projections, Hilbert space - Reflexivity, Hilbert space - Bounded operators, Hilbert space - Unbounded operators

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Hilbert spaces were named after David Hilbert, who studied them in the context of integral equations. The origin of the designation "der abstrakte Hilbertsche Raum" is John von Neumann in his famous work on unbounded Hermitian operators published in 1929. Von Neumann was perhaps the mathematician who most clearly recognized their importance as a result of his seminal work on the foundations of quantum mechanics begun with Hilbert and Lothar (Wolfgang) Nordheim and continued with Eugene Wigner. The name "Hilbert space" was soon adopted by oth ...

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Hilbert space, Hilbert space - Introduction, Hilbert space - Definition, Hilbert space - Examples, Hilbert space - Euclidean spaces, Hilbert space - Sequence spaces, Hilbert space - Lebesgue spaces, Hilbert space - Sobolev spaces, Hilbert space - Operations on Hilbert spaces, Hilbert space - Bases, Hilbert space - Orthogonal complements and projections, Hilbert space - Reflexivity, Hilbert space - Bounded operators, Hilbert space - Unbounded operators

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The polar decomposition of any bounded linear operator A between complex Hilbert spaces is a canonical factorization as the product of a partial isometry and a non-negative self-adjoint operator. Viewing complex numbers as the bounded linear mappings acting by multiplication on the complex numbers, this factorizes any bounded linear mapping z : C → C uniquely as a product of the non-negative self-adjoint operator r and the unitary operator See also:

Polar decomposition, Polar decomposition - Matrix polar decomposition, Polar decomposition - Hilbert space, Polar decomposition - Unbounded operators

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David Hilbert (January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. He established his reputation in a broad range of fields including invariant theory, the axiomization of geometry and the foundations of functional analysis. Later in life, he became a world leader in mathematics, exemplified by his presentation, in 1900, of a set of probl ...

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David Hilbert - Major contributions David Hilbert - Miscellaneous talks essays and contributions
David Hilbert - Hilbert's program
David Hilbert - Later years

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In common usage, the dimensions (from Latin "measured out") of an object are the parameters or measurements required to define its shape and size, that is, usually, its height, width, and length. In mathematics, the dimensions of a space are the parameters required to describe a particular object in this space. The dimension of a space is the number of these parameters. For example, locating a city on the Earth requires two parameters: longitude and latitude; the corresponding space has therefore two dimensions an ...

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Dimension - Physical dimensions
Dimension - Spatial dimensions Dimension - Time Dimension - Additional dimensions Dimension - Units
Dimension - Mathematical dimensions
Dimension - Hamel dimension Dimension - Manifolds Dimension - Lebesgue covering dimension Dimension - Inductive dimension Dimension - Hausdorff dimension Dimension - Hilbert spaces Dimension - Krull dimension of commutative rings
Dimension - Science fiction Dimension - Anaglyph Dimension - 3-D film
Dimension - Modern 3-D films
Dimension - More dimensions
Dimension - Degrees of freedom Dimension - Other

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In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. Many of the infinite-dimensional function spaces studied in functional analysis are examples of Banach spaces. Banach space - Definition. Banach spaces are defined as complete normed vector spaces. This means that a Banach space is a vector space V over the real or complex numbers with a norm ||.|| such that every Cauchy sequence (with respect to the metric dIncluding:

Banach space - Definition Banach space - Examples Banach space - Linear operators Banach space - Dual space Banach space - Relationship to Hilbert spaces Banach space - Derivatives Banach space - Generalizations Banach space - Literature

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In mathematics and physics, continuous spectrum is, roughly speaking, a non-countable set of eigenvalues of an operator. An operator acting on a Hilbert space is said to have a continuous spectrum if its eigenvalues can be changed continuously. If the spectrum of an operator is not continuous, we say that it is has discrete spectrum. Some of the basic questions in spectral theory are to characterise the discrete spectrum and purely continuous spectrum, just as a measure, such as a probability measure, can typically ...

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It's sometimes convenient to think of the Hilbert cube as a metric space, indeed as a specific subset of a Hilbert space with countably infinite dimension. For these purposes, it's best not to think of it as a product of copies of [0,1], but instead as [0,1] × [0,1/2] × [0,1/3] × ···; for topological properties, this makes no difference. That is, an element of the Hilbert cube is an infinite sequence (xn) that satisfies

Hilbert cube, Hilbert cube - Definition, Hilbert cube - The Hilbert cube as a metric space, Hilbert cube - Properties

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The obvious way to define the tensor product of two Banach spaces A and B is to copy the method for Hilbert spaces: define a norm on the algebraic tensor product, then take the completion in this norm. The problem is that there is more than one natural way to define a norm on the tensor product. A cross norm p on the algebraic tensor product of A and B is a norm satisfying the conditions p(a⊗b) = ||a|| || ...

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Topological tensor product, Topological tensor product - Tensor products of Hilbert spaces, Topological tensor product - Cross norms and tensor products of Banach spaces, Topological tensor product - Tensor products of locally convex topological vector spaces

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Suppose H is a Hilbert space, with inner product <.,.>. Consider a continuous linear operator A : H → H (this is the same as a bounded operator). Using the Riesz representation theorem, one can show that there exists a unique continuous linear operator A* : H → H with the following property: This operator A* is the adjoint of A. ...

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Hermitian adjoint, Hermitian adjoint - Definition for bounded operators, Hermitian adjoint - Properties, Hermitian adjoint - Hermitian operators, Hermitian adjoint - Adjoints of unbounded operators, Hermitian adjoint - Other adjoints

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Hilbert solved several important problems in the theory of invariants. Hilbert's basis theorem solved the principal problem in nineteenth century invariant theory by showing that any form of a given number of variables and of a given degree has a finite, yet complete system of independent rational integral invariants and covariants. He also unified the field of algebraic number theory with his 1897 treatis ...

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David Hilbert, David Hilbert - Major contributions, David Hilbert - Miscellaneous talks essays and contributions, David Hilbert - Hilbert's program, David Hilbert - Later years, David Hilbert - Notes

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Hilbert solved several important problems in the theory of invariants. Hilbert's basis theorem solved the principal problem in nineteenth century invariant theory by showing that any form of a given number of variables and of a given degree has a finite, yet complete system of independent rational integral invariants and covariants. He also unified the field of algebraic number theory with his 1897 treatis ...

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David Hilbert, David Hilbert - Major contributions, David Hilbert - Miscellaneous talks essays and contributions, David Hilbert - Hilbert's program, David Hilbert - Later years

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The algebraic tensor product of two Hilbert spaces A and B has a natural positive definite sesquilinear form induced by the sesquilinear forms of A and B. So in particular it has a natural positive definite quadratic form, and the corresponding completion is a Hilbert space A⊗B, called the (Hilbert space) tensor product of A and B. If the vectors ai and bj run through orthonormal bases of A and B, then the vectors a< ...

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Topological tensor product, Topological tensor product - Tensor products of Hilbert spaces, Topological tensor product - Cross norms and tensor products of Banach spaces, Topological tensor product - Tensor products of locally convex topological vector spaces

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He put forth a most influential list of 23 unsolved problems at the International Congress of Mathematicians in Paris in 1900. This is generally reckoned the most successful and deeply considered compilation of open problems ever to be produced by an individual mathematician. Additionally, Hilbert's work anticipated and assisted several advances in the mathematical formulation of quantum mechanics. These include his introduction of Hilbert space, and Hermann Weyl's proof of the mathematical equivalence o ...

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David Hilbert, David Hilbert - Major contributions, David Hilbert - Miscellaneous talks essays and contributions, David Hilbert - Hilbert's program, David Hilbert - Later years, David Hilbert - Notes

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He put forth a most influential list of 23 unsolved problems at the International Congress of Mathematicians in Paris in 1900. This is generally reckoned the most successful and deeply considered compilation of open problems ever to be produced by an individual mathematician. Additionally, Hilbert's work anticipated and assisted several advances in the mathematical formulation of quantum mechanics. These include his introduction of Hilbert space, and Hermann Weyl's proof of the mathematical equivalence o ...

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David Hilbert, David Hilbert - Major contributions, David Hilbert - Miscellaneous talks essays and contributions, David Hilbert - Hilbert's program, David Hilbert - Later years

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The topologies of locally convex topological vector spaces A and B are given by families of seminorms. For each choice of seminorm on A and on B we can define the corresponding family of cross norms on the algebraic tensor product A⊗B, and by choosing one cross norm from each family we get some cross norms on A⊗B, defining a topology. There are in general an enormous number of ways to do this. The two most important ways are to take all the projective cross norms, or all the injecti ...

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Topological tensor product, Topological tensor product - Tensor products of Hilbert spaces, Topological tensor product - Cross norms and tensor products of Banach spaces, Topological tensor product - Tensor products of locally convex topological vector spaces

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This theorem establishes an important connection between a Hilbert space and its dual space: if the ground field is the real numbers, the two are isometrically isomorphic; if the ground field is the complex numbers, the two are isometrically anti-isomorphic. The theorem is the justification for the bra-ket notation popular in the mathematical treatment of quantum mechanics. The (anti-) isomorphism is a particular natural one as will be described next. Let H be a Hilbert space, and let H ' denote its dual space, cons ...

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Riesz representation theorem, Riesz representation theorem - The Hilbert space representation theorem, Riesz representation theorem - The representation theorem for linear functionals on CcX, Riesz representation theorem - The representation theorem for the dual of C0X, Riesz representation theorem - External link

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There are many topologies that can be defined on L(H) besides the ones used above. These topologies are all locally convex, which implies that they are defined by a family of seminorms. The Banach space L(H) has a (unique) predual L(H)*, consisting of the trace class operators, whose dual is L(H). The seminorm pw(x) for w positive in t ...

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Operator topology, Operator topology - Introduction, Operator topology - List of topologies on LH, Operator topology - Relations between the topologies, Operator topology - Which topology should I use?

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Throughout, let K stand for one of the fields R or C. The familiar Euclidean spaces Kn, where the Euclidean norm of x = (x1, ..., xn) is given by ||x|| = (∑ |xi|2)1/2, are Banach spaces. The space of all continuous functions f : [a, b] → K defined on a closed interval [a, b] becomes a Banach space if we define the norm of such a f ...

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Banach space, Banach space - Definition, Banach space - Examples, Banach space - Linear operators, Banach space - Dual space, Banach space - Relationship to Hilbert spaces, Banach space - Derivatives, Banach space - Generalizations, Banach space - Literature

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