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Distinguished Space
In functional analysis and related areas of mathematics, distinguished spaces are topological vector spaces (TVSs) having the property that weak-* bounded subsets of their biduals (that is, the strong dual space of their strong dual space) are contained in the weak-* closure of some bounded subset of the bidual. Definition Suppose that X is a locally convex space and let X^ and X^_b denote the strong dual of X (that is, the continuous dual space of X endowed with the strong dual topology). Let X^ denote the continuous dual space of X^_b and let X^_b denote the strong dual of X^_b. Let X^_ denote X^ endowed with the weak-* topology induced by X^, where this topology is denoted by \sigma\left(X^, X^\right) (that is, the topology of pointwise convergence on X^). We say that a subset W of X^ is \sigma\left(X^, X^\right)-bounded if it is a bounded subset of X^_ and we call the closure of W in the TVS X^_ the \sigma\left(X^, X^\right)-closure of W. If B is a subset of X then ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word '' functional'' as a noun goes back to the calculus of variations, implying a function whose argument is a function. The term was first used in Hadamard's 1910 book on that subject. However, the general concept of a functional had previously been introduced in 1887 by the I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Convex Topological Vector Space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals. Fréchet spaces are locally convex spaces that are completely metrizable (with a choice of complete metric). They are generalizations of Banach spaces, which are complete vector spaces with respect to a metric generated by a norm. History Metrizable topologies on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) such that the topology induced by d is \mathcal. Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable. Properties Metrizable spaces inherit all topological properties from metric spaces. For example, they are Hausdorff paracompact spaces (and hence normal and Tychonoff) and first-countable. However, some properties of the metric, such as completeness, cannot be said to be inherited. This is also true of other structures linked to the metric. A metrizable uniform space, for example, may have a different set of contraction maps than a metric space to which it is homeomorphic. Metrization theorems One of the first widely recognized metrization theorems was . This states that every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lp Space
In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourbaki group they were first introduced by Frigyes Riesz . spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines. Applications Statistics In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, are defined in terms of metrics, and measures of central tendency can be characterized as solutions to variational problems. In penalized regression, "L1 penalty" and "L2 penalty" refer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separable Space
In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence \_^ of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence. Like the other axioms of countability, separability is a "limitation on size", not necessarily in terms of cardinality (though, in the presence of the Hausdorff axiom, this does turn out to be the case; see below) but in a more subtle topological sense. In particular, every continuous function on a separable space whose image is a subset of a Hausdorff space is determined by its values on the countable dense subset. Contrast separability with the related notion of second countability, which is in general stronger but equivalent on the class of metrizable spaces. First examples Any topological space that is itself finite or countably infinite is separable, for the whole space is a countable dense subset of itself. An impo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semi-reflexive
In the area of mathematics known as functional analysis, a semi-reflexive space is a locally convex topological vector space (TVS) ''X'' such that the canonical evaluation map from ''X'' into its bidual (which is the strong dual of the strong dual of ''X'') is bijective. If this map is also an isomorphism of TVSs then it is called reflexive. Semi-reflexive spaces play an important role in the general theory of locally convex TVSs. Since a normable TVS is semi-reflexive if and only if it is reflexive, the concept of semi-reflexivity is primarily used with TVSs that are not normable. Definition and notation Brief definition Suppose that is a topological vector space (TVS) over the field \mathbb (which is either the real or complex numbers) whose continuous dual space, X^, separates points on (i.e. for any x \in X there exists some x^ \in X^ such that x^(x) \neq 0). Let X^_b and X^_ both denote the strong dual of , which is the vector space X^ of continuous linear func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term " Fréchet space." Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a complete n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H-space
In mathematics, an H-space is a homotopy-theoretic version of a generalization of the notion of topological group, in which the axioms on associativity and inverses are removed. Definition An H-space consists of a topological space , together with an element of and a continuous map , such that and the maps and are both homotopic to the identity map through maps sending to . This may be thought of as a pointed topological space together with a continuous multiplication for which the basepoint is an identity element up to basepoint-preserving homotopy. One says that a topological space is an H-space if there exists and such that the triple is an H-space as in the above definition. Alternatively, an H-space may be defined without requiring homotopies to fix the basepoint , or by requiring to be an exact identity, without any consideration of homotopy. In the case of a CW complex, all three of these definitions are in fact equivalent. Examples and properties The standard ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasibarrelled Space
In functional analysis and related areas of mathematics, quasibarrelled spaces are topological vector spaces (TVS) for which every bornivorous barrelled set in the space is a Neighbourhood (topology), neighbourhood of the origin. Quasibarrelled spaces are studied because they are a weakening of the defining condition of barrelled spaces, for which a form of the Banach–Steinhaus theorem holds. Definition A subset B of a topological vector space (TVS) X is called bornivorous if it absorbs all bounded subsets of X; that is, if for each bounded subset S of X, there exists some scalar r such that S \subseteq r B. A barrelled set or a barrel in a TVS is a Set (mathematics), set which is Convex set, convex, Balanced set, balanced, Absorbing set, absorbing and Closed set, closed. A quasibarrelled space is a TVS for which every bornivorous barrelled set in the space is a Neighbourhood (topology), neighbourhood of the origin. Properties A locally convex Hausdorff quasibarrelled sp ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LF Space
In mathematics, an ''LF''-space, also written (''LF'')-space, is a topological vector space (TVS) ''X'' that is a locally convex inductive limit of a countable inductive system (X_n, i_) of Fréchet spaces. This means that ''X'' is a direct limit of a direct system (X_n, i_) in the category of locally convex topological vector spaces and each X_n is a Fréchet space. The name ''LF'' stands for Limit of Fréchet spaces. If each of the bonding maps i_ is an embedding of TVSs then the ''LF''-space is called a strict ''LF''-space. This means that the subspace topology induced on by is identical to the original topology on . Some authors (e.g. Schaefer) define the term "''LF''-space" to mean "strict ''LF''-space," so when reading mathematical literature, it is recommended to always check how ''LF''-space is defined. Definition Inductive/final/direct limit topology Throughout, it is assumed that * \mathcal is either the category of topological spaces or some subcategory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semi-reflexive Space
In the area of mathematics known as functional analysis, a semi-reflexive space is a locally convex topological vector space (TVS) ''X'' such that the canonical evaluation map from ''X'' into its bidual (which is the strong dual of the strong dual of ''X'') is bijective. If this map is also an isomorphism of TVSs then it is called reflexive. Semi-reflexive spaces play an important role in the general theory of locally convex TVSs. Since a normable TVS is semi-reflexive if and only if it is reflexive, the concept of semi-reflexivity is primarily used with TVSs that are not normable. Definition and notation Brief definition Suppose that is a topological vector space (TVS) over the field \mathbb (which is either the real or complex numbers) whose continuous dual space, X^, separates points on (i.e. for any x \in X there exists some x^ \in X^ such that x^(x) \neq 0). Let X^_b and X^_ both denote the strong dual of , which is the vector space X^ of continuous linear func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
