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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] |
Strong Topology
In mathematics, a strong topology is a topology which is stronger than some other "default" topology. This term is used to describe different topologies depending on context, and it may refer to: * the final topology on the disjoint union * the topology arising from a norm * the strong operator topology * the strong topology (polar topology), which subsumes all topologies above. A topology τ is stronger than a topology σ (is a finer topology) if τ contains all the open sets of σ. In algebraic geometry, it usually means the topology of an algebraic variety as complex manifold or subspace of complex projective space, as opposed to the Zariski topology (which is rarely even a Hausdorff space). See also * Weak topology In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ... {{sia, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normable Space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" in the real (physical) world. A norm is a real-valued function defined on the vector space that is commonly denoted x\mapsto \, x\, , and has the following properties: #It is nonnegative, meaning that \, x\, \geq 0 for every vector x. #It is positive on nonzero vectors, that is, \, x\, = 0 \text x = 0. # For every vector x, and every scalar \alpha, \, \alpha x\, = , \alpha, \, \, x\, . # The triangle inequality holds; that is, for every vectors x and y, \, x+y\, \leq \, x\, + \, y\, . A norm induces a distance, called its , by the formula d(x,y) = \, y-x\, . which makes any normed vector space into a metric space and a topological vector space. If this metric space is complete then the normed space is a Banach space. Every normed ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Disk
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 su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual System
In mathematics, a dual system, dual pair, or duality over a field \mathbb is a triple (X, Y, b) consisting of two vector spaces X and Y over \mathbb and a non-degenerate bilinear map b : X \times Y \to \mathbb. Duality theory, the study of dual systems, is part of functional analysis. According to Helmut H. Schaefer, "the study of a locally convex space in terms of its dual is the central part of the modern theory of topological vector spaces, for it provides the deepest and most beautiful results of the subject." Definition, notation, and conventions ;Pairings A or pair over a field \mathbb is a triple (X, Y, b), which may also be denoted by b(X, Y), consisting of two vector spaces X and Y over \mathbb (which this article assumes is either the real numbers or the complex numbers \Complex) and a bilinear map b : X \times Y \to \mathbb, which is called the bilinear map associated with the pairing or simply the pairing's map/bilinear form. For every x \in X, define \begin b ... [...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] |
Reflexive Space
In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space (and so in particular, every Banach space) X is reflexive if and only if the canonical evaluation map from X into its bidual is surjective; in this case the normed space is necessarily also a Banach space. In 1951, R. C. James discovered a Banach space, now known as James' space, that is reflexive but is nevertheless isometrically isomorphic to its bidual (any such isomorphism is thus necessarily the canonical evaluation map). Reflexive spaces play an important role in the general theory of locally convex TVSs and in the theory of Banach spaces in particular. Hilbert spaces are prominent examples of reflexive Banach spaces. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
DF-space
In the field of functional analysis, DF-spaces, also written (''DF'')-spaces are locally convex topological vector space having a property that is shared by locally convex metrizable topological vector spaces. They play a considerable part in the theory of topological tensor products. DF-spaces were first defined by Alexander Grothendieck and studied in detail by him in . Grothendieck was led to introduce these spaces by the following property of strong duals of metrizable spaces: If X is a metrizable locally convex space and V_1, V_2, \ldots is a sequence of convex 0-neighborhoods in X^_b such that V := \cap_ V_i absorbs every strongly bounded set, then V is a 0-neighborhood in X^_b (where X^_b is the continuous dual space of X endowed with the strong dual topology). Definition A locally convex topological vector space (TVS) X is a DF-space, also written (''DF'')-space, if # X is a countably quasi-barrelled space (i.e. every strongly bounded countable union of equicontinuo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fréchet Space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the metric induced by the norm). All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically Banach spaces. A Fréchet space X is defined to be a locally convex metrizable topological vector space (TVS) that is complete as a TVS, meaning that every Cauchy sequence in X converges to some point in X (see footnote for more details).Here "Cauchy" means Cauchy with respect to the canonical uniformity that every TVS possess. That is, a sequence x_ = \left(x_m\right)_^ in a TVS X is Cauchy if and only if for all neighborhoods U of the origin in X, x_m - x_n \in U whenever m and n are sufficiently large. Note that this definition of a Cau ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Null Sequence
As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1." In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the \lim symbol (e.g., \lim_a_n).Courant (1961), p. 29. If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests. Limits can be defined in any metric or topological space, but are usually first encountered in the real numbers. History The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involve limiting processes. Leucippus, Democritus, Antiphon, Eudoxus, and Archimedes developed the method of exhaustion, which uses an infinite sequence of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolutely Convergent
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said to converge absolutely if \textstyle\sum_^\infty \left, a_n\ = L for some real number \textstyle L. Similarly, an improper integral of a function, \textstyle\int_0^\infty f(x)\,dx, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if \textstyle\int_0^\infty , f(x), dx = L. Absolute convergence is important for the study of infinite series because its definition is strong enough to have properties of finite sums that not all convergent series possess - a convergent series that is not absolutely convergent is called conditionally convergent, while absolutely convergent series behave "nicely". For instance, rearrangements do not change the value of the sum. This is not true for condi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
F-space
In functional analysis, an F-space is a vector space X over the real or complex numbers together with a metric d : X \times X \to \R such that # Scalar multiplication in X is continuous with respect to d and the standard metric on \R or \Complex. # Addition in X is continuous with respect to d. # The metric is translation-invariant; that is, d(x + a, y + a) = d(x, y) for all x, y, a \in X. # The metric space (X, d) is complete. The operation x \mapsto \, x\, := d(0, x) is called an F-norm, although in general an F-norm is not required to be homogeneous. By translation-invariance, the metric is recoverable from the F-norm. Thus, a real or complex F-space is equivalently a real or complex vector space equipped with a complete F-norm. Some authors use the term rather than , but usually the term "Fréchet space" is reserved for locally convex F-spaces. Some other authors use the term "F-space" as a synonym of "Fréchet space", by which they mean a locally convex complete metri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
