Definitions

1. Hilbert spaces were named after David Hilbert, who studied them in the context of integral equations.
2. Hilbert spaces are studied in the branch of mathematics called functional analysis. (Web site)
3. Hilbert spaces are central to many applications, from quantum mechanics to stochastic calculus.
4. Hilbert spaces are very important for quantum theory. (Web site)
5. Hilbert spaces are of crucial importance in the mathematical formulation of quantum mechanics.

Phase Spaces

1. We finally generalizes our analysis to Hilbert spaces of prime dimensions d and their associated d*d phase spaces. (Web site)

Important Inner Product Spaces

1. The most important inner product spaces are the ones which are complete with respect to this metric; they are called Hilbert spaces. (Web site)

Fréchet Spaces

1. Hilbert spaces, Banach spaces or Fréchet spaces. (Web site)

Normed Linear Spaces

1. The most important alternatives are orthogonal bases on Hilbert spaces, Schauder bases and Markushevich bases on normed linear spaces. (Web site)

Hilbert Space Tensor Product

1. The Hilbert space tensor product of two Hilbert spaces is the completion of their algebraic tensor product. (Web site)

Symmetric Monoidal Functor

1. A chain field theory is defined to be a symmetric monoidal functor from this category into the category of Hilbert spaces. (Web site)

Infinite-Dimensional Topological Vector Spaces

1. Of all the infinite-dimensional topological vector spaces, the Hilbert spaces are the most "well-behaved" and the closest to the finite-dimensional spaces.

Analogous Spectral Theorem

1. There is also an analogous spectral theorem for bounded normal operators on Hilbert spaces. (Web site)

Function Spaces

1. Hilbert spaces allow simple geometric concepts, like projection and change of basis to be applied to infinite dimensional spaces, such as function spaces.

Quantum Mechanical Systems

1. Quantum mechanical systems are represented by Hilbert spaces, which are anti - isomorphic to their own dual spaces.

General Banach Spaces

1. General Banach spaces are more complicated, and cannot be classified in such a simple manner as Hilbert spaces. (Web site)

Banach

1. The second part deals with the two basic spaces of analysis, Banach and Hilbert spaces.

Infinite-Dimensional Hilbert Spaces

1. Some but not all of the above generalize to normal operators on infinite-dimensional Hilbert spaces. (Web site)
2. Infinite-dimensional Hilbert spaces are central to the subject.

Reproducing Kernel Hilbert Spaces

1. The book first rigorously develops the theory of reproducing kernel Hilbert spaces.

Infinite Dimensional Hilbert Spaces

1. The partial trace generalizes to operators on infinite dimensional Hilbert spaces.

Finite Dimensional Hilbert Spaces

1. The category FdHilb of finite dimensional Hilbert spaces and linear maps. (Web site)

Section

1. Small complaint about the section on Hilbert spaces, in the sketch of the proof that parallelogram identity implies inner-product norm.

Discussing

1. They would hold for normed spaces, and perhps the intention was to continue discussing only Hilbert spaces.

Elements

1. Hilbert spaces, elements of spectral theory.

Context

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

Topics

1. Topics covered in Shilov: Function spaces, L^p-spaces, Hilbert spaces, and linear operators; the standard Banach, and Hahn-Banach theorems.
2. Other topics include Banach and Hilbert spaces, multinormal and uniform spaces, and the Riesz-Dunford holomorphic functional calculus.
3. The topics covered include: Hilbert spaces, fundamental properties of bounded linear operators, and further developments of bounded linear operators. (Web site)

Quantum

1. QM on real Hilbert spaces) still satisfy PURIFY. Finally I will address the problem on how to prove that a test-theory is quantum.

Formulation

1. Another area where this formulation is used is in Hilbert spaces. (Web site)
2. We will exclusively deal with infinite dimensional Hilbert spaces and will not attempt to include the simpler finite dimensional case in our formulation. (Web site)

Geometry

1. Lp-spaces, Riesz-Fischer theorem, geometry of Hilbert spaces.

Extended

1. The construction may also be extended to cover Banach spaces and Hilbert spaces. (Web site)
2. The concept of eigenvectors can be extended to linear operators acting on infinite dimensional Hilbert spaces or Banach spaces. (Web site)

Vector Space

1. The vector space need not be finite-dimensional; thus, for example, there is the theory of projective Hilbert spaces.

Subspaces

1. In infinite-dimensional Hilbert spaces, some subspaces are not closed, but all orthogonal complements are closed.

Special Case

1. To repeat that I'm repeating myself, Hilbert spaces are a special case of an inner product space.

Manifold

1. On the other hand, a Hilbert manifold is a special case of a Banach manifold in which the manifold is locally modelled on Hilbert spaces.

Chapter

1. Sesquilinear forms in Hilbert spaces are considered in detail (chapter 6), analytic and asymptotic perturbation theory is described (chapter 7 and 8). (Web site)
2. A chapter is devoted to the exciting interplay between numerical range conditions, similarity problems and functional calculus on Hilbert spaces. (Web site)

Chapters

1. Also, the book only touches on functional analysis in the two chapters on Hilbert spaces (where they assume all Hilbert spaces are separable).
2. The chapters on Banach and Hilbert spaces should be of special interest to people who wish to study the solution of variational boundary value problems. (Web site)

Terms

1. An example of such a formal definition in terms of Lie group representations on Hilbert spaces of quantum states and operators is provided next.

Definition

1. The definition of the Friedrichs extension is based on the theory of closed positive forms on Hilbert spaces.
2. As a complete normed space, Hilbert spaces are by definition also Banach spaces. (Web site)

Distributions

1. The techniques of modern analysis, such as distributions and Hilbert spaces, are used wherever appropriate to illuminate these long-studied topics. (Web site)

David Hilbert

1. Complete inner product spaces are known as Hilbert spaces, in honor of David Hilbert.

Complex

1. The above spectral theorem holds for real or complex Hilbert spaces. (Web site)
2. In this article we assume that Hilbert spaces are complex.

Integrals

1. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras.
2. The spectral multiplicity theorem can be reformulated using the language of direct integrals of Hilbert spaces: Theorem.

Algebras

1. Algebras of upper triangular matrices have a natural generalization in functional analysis which yields nest algebras on Hilbert spaces. (Web site)

Normal Operators

1. There is also an analogous spectral theorem for normal operators on Hilbert spaces.
2. The author returns to Hilbert spaces in chapter 11, wherein he introduces the highly important class of normal operators.
3. Some applications of the functional-representations of normal operators in Hilbert spaces. (Web site)

Bounded

1. We also show extensions valid for bounded and also unbounded operators in Hilbert spaces, which allow the development of a functional calculus. (Web site)
2. Let H and K be Hilbert spaces and for each let be a bounded but not necessarily compact linear map with A(z) analytic on a region z < a.

Important

1. An important object of study in functional analysis are the continuous linear operators defined on Banach and Hilbert spaces.
2. SCALAR OR DOT PRODUCT 13 spaces, the Hilbert spaces, which are, important in modern quantum theory.

Lie Groups

1. Representations of these Lie groups (after -deformations, in general) on Hilbert spaces are called quantizations.

Examples

1. Examples of finite-dimensional Hilbert spaces include 1. (Web site)

Set

1. In fact, by choosing a Hilbert basis, one sees that all Hilbert spaces are isometric to ℓ 2(E), where E is a set with an appropriate cardinality. (Web site)
2. In this brief note I would like to set out the abstract theory of such higher order Hilbert spaces.

Concept

1. Even more general is the concept of adjoint operator for operators on (possibly infinite-dimensional) complex Hilbert spaces. (Web site)
2. The concept of trace of a matrix is generalised to the trace class of bounded linear operators on Hilbert spaces.
3. The concept of eigenvectors can be extended to linear operators acting on infinite-dimensional Hilbert spaces or Banach spaces.

Topology

1. An important object of study in functional analysis are the continuous function (topology) linear transformation defined on Banach and Hilbert spaces.

Categories

1. Mathematics > Mathematical Analysis > Functional Analysis > Banach Spaces
2. Mathematics > Mathematical Analysis > Functional Analysis > Hilbert Space
3. Science > Mathematics > Mathematical Analysis > Functional Analysis
4. Tensor Product
5. Operators

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