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Schwartz Kernel Theorem
In mathematics, the Schwartz kernel theorem is a foundational result in the theory of generalized functions, published by Laurent Schwartz in 1952. It states, in broad terms, that the generalized functions introduced by Schwartz (Schwartz distributions) have a two-variable theory that includes all reasonable bilinear forms on the space \mathcal of test functions. The space \mathcal itself consists of smooth functions of compact support. Statement of the theorem Let X and Y be open sets in \mathbb^n. Every distribution k \in \mathcal'(X \times Y) defines a continuous linear map K \colon \mathcal(Y) \to \mathcal'(X) such that for every u \in \mathcal(X), v \in \mathcal(Y). Conversely, for every such continuous linear map K there exists one and only one distribution k \in \mathcal'(X \times Y) such that () holds. The distribution k is the kernel of the map K. Note Given a distribution k \in \mathcal'(X \times Y) one can always write the linear map K informally as :Kv = \int_ k(\c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Support Of A Function
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used very widely in mathematical analysis. Formulation Suppose that f : X \to \R is a real-valued function whose domain is an arbitrary set X. The of f, written \operatorname(f), is the set of points in X where f is non-zero: \operatorname(f) = \. The support of f is the smallest subset of X with the property that f is zero on the subset's complement. If f(x) = 0 for all but a finite number of points x \in X, then f is said to have . If the set X has an additional structure (for example, a topology), then the support of f is defined in an analogous way as the smallest subset of X of an appropriate type such that f vanishes in an appropriate sense on its complement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Functions
In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing discrete physical phenomena such as point charges. They are applied extensively, especially in physics and engineering. A common feature of some of the approaches is that they build on operator aspects of everyday, numerical functions. The early history is connected with some ideas on operational calculus, and more contemporary developments in certain directions are closely related to ideas of Mikio Sato, on what he calls algebraic analysis. Important influences on the subject have been the technical requirements of theories of partial differential equations, and group representation theory. Some early history In the mathematics of the nineteenth century, aspects of generalized function theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace Class
In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a Trace (linear algebra), trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace-class operators generalizes the trace of matrices studied in linear algebra. All trace-class operators are Compact operator, compact operators. In quantum mechanics, Mixed state (physics), mixed states are described by Density matrix, density matrices, which are certain trace class operators. Trace-class operators are essentially the same as nuclear operators, though many authors reserve the term "trace-class operator" for the special case of nuclear operators on Hilbert spaces and use the term "nuclear operator" in more general topological vector spaces (such as Banach spaces). Note that the trace operator studied in partial differential equations is an unrelated concept. Definition Suppose H is a Hilbert s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rigged Hilbert Space
In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study spectral theory in the broad sense. They bring together the 'bound state' (eigenvector) and ' continuous spectrum', in one place. Motivation A function such as the canonical homomorphism of the real line into the complex plane : x \mapsto e^ , is an eigenfunction of the differential operator :-i\frac on the real line R, but isn't square-integrable for the usual Borel measure on R. To properly consider this function as an eigenfunction requires some way of stepping outside the strict confines of the Hilbert space theory. This was supplied by the apparatus of Schwartz distributions, and a ''generalized eigenfunction'' theory was developed in the years after 1950. Functional analysis approach The concept of rigged Hilbert space places ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Tensor Product
The strongest locally convex topological vector space (TVS) topology on X \otimes Y, the tensor product of two locally convex TVSs, making the canonical map \cdot \otimes \cdot : X \times Y \to X \otimes Y (defined by sending (x, y) \in X \times Y to x \otimes y) continuous is called the projective topology or the π-topology. When X \otimes Y is endowed with this topology then it is denoted by X \otimes_ Y and called the projective tensor product of X and Y. Preliminaries Throughout let X, Y, and Z be topological vector spaces and L : X \to Y be a linear map. * L : X \to Y is a topological homomorphism or homomorphism, if it is linear, continuous, and L : X \to \operatorname L is an open map, where \operatorname L, the image of L, has the subspace topology induced by Y. ** If S \subseteq X is a subspace of X then both the quotient map X \to X / S and the canonical injection S \to X are homomorphisms. In particular, any linear map L : X \to Y can be canonically decomposed as f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nuclear Operator
In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spaces (TVSs). Preliminaries and notation Throughout let ''X'',''Y'', and ''Z'' be topological vector spaces (TVSs) and ''L'' : ''X'' → ''Y'' be a linear operator (no assumption of continuity is made unless otherwise stated). * The projective tensor product of two locally convex TVSs ''X'' and ''Y'' is denoted by X \otimes_ Y and the completion of this space will be denoted by X \widehat_ Y. * ''L'' : ''X'' → ''Y'' is a topological homomorphism or homomorphism, if it is linear, continuous, and L : X \to \operatorname L is an open map, where \operatorname L, the image of ''L'', has the subspace topology induced by ''Y''. ** If ''S'' is a subspace of ''X'' then both the quotient map ''X'' → ''X''/''S'' and the canonical injection ''S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Injective Tensor Product
In mathematics, the injective tensor product of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so its completion is called the . Injective tensor products have applications outside of nuclear spaces. In particular, as described below, up to TVS-isomorphism, many TVSs that are defined for real or complex valued functions, for instance, the Schwartz space or the space of continuously differentiable functions, can be immediately extended to functions valued in a Hausdorff locally convex TVS Y with any need to extend definitions (such as "differentiable at a point") from real/complex-valued functions to Y-valued functions. Preliminaries and notation Throughout let X, Y, and Z be topological vector spaces and L : X \to Y be a linear map. * L : X \to Y is a topological homomorphism or homomorphism, if it is linear, continuous, and L : X \to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fredholm Kernel
In mathematics, a Fredholm kernel is a certain type of a kernel on a Banach space, associated with nuclear operators on the Banach space. They are an abstraction of the idea of the Fredholm integral equation and the Fredholm operator, and are one of the objects of study in Fredholm theory. Fredholm kernels are named in honour of Erik Ivar Fredholm. Much of the abstract theory of Fredholm kernels was developed by Alexander Grothendieck and published in 1955. Definition Let ''B'' be an arbitrary Banach space, and let ''B''* be its dual, that is, the space of bounded linear functionals on ''B''. The tensor product B^*\otimes B has a completion under the norm :\Vert X \Vert_\pi = \inf \sum_ \Vert e^*_i\Vert \Vert e_i \Vert where the infimum is taken over all finite representations :X=\sum_ e^*_i \otimes e_i \in B^*\otimes B The completion, under this norm, is often denoted as :B^* \widehat_\pi B and is called the projective topological tensor product. The elements of this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nuclear Space
In mathematics, nuclear spaces are topological vector space, topological vector spaces that can be viewed as a generalization of finite dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces are however quite different from Hilbert spaces, another generalization of finite dimensional Euclidean spaces. They were introduced by Alexander Grothendieck. The topology on nuclear spaces can be defined by a family of seminorms whose Unit sphere#Unit balls in normed vector spaces, unit balls decrease rapidly in size. Vector spaces whose elements are "smooth" in some sense tend to be nuclear spaces; a typical example of a nuclear space is the set of smooth functions on a compact manifold. All finite-dimensional vector spaces are nuclear. There are no Banach space, Banach spaces that are nuclear, except for the finite-dimensional ones. In practice a sort of converse to this is often true: if a "naturally occurring" topological vector space is a Banach space, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nuclear Space
In mathematics, nuclear spaces are topological vector space, topological vector spaces that can be viewed as a generalization of finite dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces are however quite different from Hilbert spaces, another generalization of finite dimensional Euclidean spaces. They were introduced by Alexander Grothendieck. The topology on nuclear spaces can be defined by a family of seminorms whose Unit sphere#Unit balls in normed vector spaces, unit balls decrease rapidly in size. Vector spaces whose elements are "smooth" in some sense tend to be nuclear spaces; a typical example of a nuclear space is the set of smooth functions on a compact manifold. All finite-dimensional vector spaces are nuclear. There are no Banach space, Banach spaces that are nuclear, except for the finite-dimensional ones. In practice a sort of converse to this is often true: if a "naturally occurring" topological vector space is a Banach space, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
