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Nuclear Spaces
In mathematics, nuclear spaces are 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 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 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, then there is a good chance that it is nuclear. Original motivation: The Schwartz ... [...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 poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nuclear Map
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Tensor Product
In mathematics, there are usually many different ways to construct a topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products (see Tensor product of Hilbert spaces), but for general Banach spaces or locally convex topological vector spaces the theory is notoriously subtle. Motivation One of the original motivations for topological tensor products \hat is the fact that tensor products of the spaces of smooth functions on \R^n do not behave as expected. There is an injection :C^\infty(\R^n) \otimes C^\infty(\R^m) \hookrightarrow C^\infty(\R^) but this is not an isomorphism. For example, the function f(x,y) = e^ cannot be expressed as a finite linear combination of smooth functions in C^\infty(\R_x)\otimes C^\infty(\R_y). We only get an isomorphism after constructing the topological tensor product; i.e., :C^\infty(\R^n) \mathop C^\infty(\R^m) \cong C^\infty(\R^) This article first de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equicontinuous
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus ''sequences'' of functions. Equicontinuity appears in the formulation of Ascoli's theorem, which states that a subset of ''C''(''X''), the space of continuous functions on a compact Hausdorff space ''X'', is compact if and only if it is closed, pointwise bounded and equicontinuous. As a corollary, a sequence in ''C''(''X'') is uniformly convergent if and only if it is equicontinuous and converges pointwise to a function (not necessarily continuous a-priori). In particular, the limit of an equicontinuous pointwise convergent sequence of continuous functions ''fn'' on either metric space or locally compact space is continuous. If, in addition, ''fn'' are holomorphic, then the limit is also holomorphic. The uniform bou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert–Schmidt Operator
In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A \colon H \to H that acts on a Hilbert space H and has finite Hilbert–Schmidt norm \, A\, ^2_ \ \stackrel\ \sum_ \, Ae_i\, ^2_H, where \ is an orthonormal basis. The index set I need not be countable. However, the sum on the right must contain at most countably many non-zero terms, to have meaning. This definition is independent of the choice of the orthonormal basis. In finite-dimensional Euclidean space, the Hilbert–Schmidt norm \, \cdot\, _\text is identical to the matrix norm#Frobenius norm, Frobenius norm. , , ·, , is well defined The Hilbert–Schmidt norm does not depend on the choice of orthonormal basis. Indeed, if \_ and \_ are such bases, then \sum_i \, Ae_i\, ^2 = \sum_ \left, \langle Ae_i, f_j\rangle \^2 = \sum_ \left, \langle e_i, A^*f_j\rangle \^2 = \sum_j\, A^* f_j\, ^2. If e_i = f_i, then \sum_i \, Ae_i\, ^2 = \sum_i\, A^* e_i\, ^2. ... [...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 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 operators. In quantum mechanics, mixed states are described by 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 space and A : H \to H a bounded linear operator on H which is non-negative (I.e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subbase
In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by some authors, and there are other useful equivalent formulations of the definition; these are discussed below. Definition Let X be a topological space with topology T. A subbase of T is usually defined as a subcollection B of T satisfying one of the two following equivalent conditions: #The subcollection B ''generates'' the topology T. This means that T is the smallest topology containing B: any topology T^\prime on X containing B must also contain T. #The collection of open sets consisting of all finite intersections of elements of B, together with the set X, forms a basis for T. This means that every proper open set in T can be written as a union of finite intersections of elements of B. Explicitly, given a point x in an open set U \ ... [...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 ' ... [...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 ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Auxiliary Normed Space
In functional analysis, two methods of constructing normed spaces from disks were systematically employed by Alexander Grothendieck to define nuclear operators and nuclear spaces. One method is used if the disk D is bounded: in this case, the auxiliary normed space is \operatorname D with norm p_D(x) := \inf_ r. The other method is used if the disk D is absorbing: in this case, the auxiliary normed space is the quotient space X / p_D^(0). If the disk is both bounded and absorbing then the two auxiliary normed spaces are canonically isomorphic (as topological vector spaces and as normed spaces). Preliminaries A subset of a vector space is called a disk and is said to be disked, absolutely convex, or convex balanced if it is convex and balanced. If C and D are subsets of a vector space X then D absorbs C if there exists a real r > 0 such that C \subseteq a Dfor any scalar a satisfying , a, \geq r. WThe set D is called absorbing in X if D absorbs \ for every x \in X. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Topological Vector Space
In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point x towards which they all get closer. The notion of "points that get progressively closer" is made rigorous by or , which are generalizations of , while "point x towards which they all get closer" means that this Cauchy net or filter converges to x. The notion of completeness for TVSs uses the theory of uniform spaces as a framework to generalize the notion of completeness for metric spaces. But unlike metric-completeness, TVS-completeness does not depend on any metric and is defined for TVSs, including those that are not metrizable or Hausdorff. Completeness is an extremely important property for a topological vector space to possess. The notions of completeness for normed spaces and metrizable TVSs, which are commonly defined in terms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Balanced Set
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, \leq 1. The balanced hull or balanced envelope of a set S is the smallest balanced set containing S. The balanced core of a subset S is the largest balanced set contained in S. Balanced sets are ubiquitous in functional analysis because every neighborhood of the origin in every topological vector space (TVS) contains a balanced neighborhood of the origin and every convex neighborhood of the origin contains a balanced convex neighborhood of the origin (even if the TVS is not locally convex). This neighborhood can also be chosen to be an open set or, alternatively, a closed set. Definition Let X be a vector space over the field \mathbb of real or complex numbers. Notation If S is a set, a is a scalar, and B \subseteq \mathbb t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
