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Encyclopedia > Topological vector space

In mathematics a topological vector space is one of the basic structures investigated in functional analysis. As the name suggests the space blends a topological structure (a uniform structure to be precise) with the algebraic concept of a vector space. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... In topology, one defines uniform spaces in order to study concepts such as uniform continuity, completeness and uniform convergence. ... Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...


The elements of topological vector spaces are typically functions, and the topology is often defined so as to capture a particular notion of convergence of sequences of functions. Partial plot of a function f. ...


Hilbert spaces and Banach spaces are well-known examples. In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...

Contents

Definition

A topological vector space X is a vector space over a topological field K (most often the real or complex numbers with their standard topologies) which is endowed with a topology such that vector addition X × XX and scalar multiplication K × XX are continuous functions. In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps R × R → R, where R × R carries the product topology. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ...


N.B. Though we do not do so here, some authors require the topology on X to be Hausdorff, and some additionally require the topology on X to be locally convex (e.g., Fréchet space). However, it is known that for a topological vector space to be Hausdorff it suffices that the space is a T1 space. In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In functional analysis, a topological vector space is called locally convex if its topology is defined by a set of convex neighborhoods of 0. ... This article deals with Fréchet spaces in functional analysis. ... The title given to this article is incorrect due to technical limitations. ...


The category of topological vector spaces over a given topological field K is commonly denoted TVSK or TVectK. The objects are the topological vector spaces over K and the morphisms are the continuous K-linear maps from one object to another. In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. ...


Examples

All normed vector spaces (and therefore all Banach spaces and Hilbert spaces) are examples of topological vector spaces. In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ...


Product vector spaces

A cartesian product of a family of topological vector spaces, when endowed with the product topology is a topological vector space. For instance, the set X of all functions f : RR. X can be identified with the product space RR and carries a natural product topology. With this topology, X becomes a topological vector space, called the space of pointwise convergence. The reason for this name is the following: if (fn) is a sequence of elements in X, then fn has limit f in X if and only if fn(x) has limit f(x) for every real number x. This space is complete, but not normable. In mathematics, the Cartesian product is a direct product of sets. ... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ... Suppose { fn } is a sequence of functions sharing the same domain in common (for the moment, we defer making precise the nature of the values of these functions, but the reader may take them to be real numbers if that makes anyone feel good). ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as...


Topological structure

A vector space is an abelian group with respect to the operation of addition, and in a topological vector space the inverse operation is always continuous (since it is the same as multiplication by −1). Hence, every topological vector space is an abelian topological group. In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ...


In particular, topological vector spaces are uniform spaces and one can thus talk about completeness, uniform convergence and uniform continuity. The vector space operations of addition and scalar multiplication are actually uniformly continuous. Because of this, every topological vector space can be completed and is thus a dense linear subspace of a complete topological vector space. In the mathematical field of topology, a uniform space is a set with a uniform structure. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. ... In mathematical analysis, a function f(x) is called uniformly continuous if, roughly speaking, small changes in the input x effect small changes in the output f(x) (continuity), and furthermore the size of the changes in f(x) depends only on the size of the changes in x but... In mathematics, the term dense has at least three different meanings. ... The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ...


Vector addition and scalar multiplication are not only continuous but even homeomorphisms which means we can construct a base for the topology and thus reconstruct the whole topology of the space from any local base around the origin. In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases... This is a glossary of some terms used in the branch of mathematics known as topology. ...


Every topological vector space has a local base of absorbing and balanced sets. In functional analysis and related areas of mathematics an absorbing set in a vector space is a set S which can be inflated to include any element of the vector space. ... In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field K with an absolute value |.|) is a set S so that for all scalars α with |α| ≤ 1 with The balanced hull or balanced envelope for a set S is...


If a topological vector space is semi-metrisable, that is the topology can be given by a semi-metric, then the semi-metric must be translation invariant. Also, a topological vector space is metrizable if and only if it is Hausdorff and has a countable local base (i.e., a neighborhood base at the origin). In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. ... In mathematics a metric or distance function is a function which defines a distance between elements of a set. ... In mathematics a metric or distance is a function which assigns a distance to elements of a set. ... A metrizable space is a topological space that is homeomorphic to a metric space. ...


A linear function between two topological vector spaces which is continuous at one point is continuous on the whole domain.


A linear functional f on a topological vector space X is continuous if and only if kernel(f) is closed in X. In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. ... In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ...


If a vector space is finite dimensional, then there is a unique Hausdorff topology on it. Thus any finite dimensional topological vector space is isomorphic to Kn. A Hausdorff topological vector space is finite-dimensional if and only if it is locally compact. Here isomorphism means that there exists a linear homeomorphism between the two spaces. In mathematics, the dimension of a vector space V is the cardinality (i. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ...


Types of topological vector spaces

Depending on the application we usually enforce additional constraints on the topological structure of the space. Below are some common topological vector spaces, roughly ordered by their niceness.

  • Locally convex topological vector spaces: here each point has a local base consisting of convex sets. By a technique known as Minkowski functionals it can be shown that a space is locally convex if and only if its topology can be defined by a family of semi-norms. Local convexity is the minimum requirement for "geometrical" arguments like the Hahn-Banach theorem.
  • Barrelled spaces: locally convex spaces where the Banach-Steinhaus theorem holds.
  • Montel space: a barrelled space where every closed and bounded set is compact
  • Bornological space: a locally convex space where the continuous linear operators to any locally convex space are exactly the bounded linear operators.
  • LF-spaces are limits of Fréchet spaces. ILH spaces are inverse limits of Hilbert spaces.
  • F-spaces are complete topological vector spaces with a translation-invariant metric. These include Lp spaces for all p > 0.
  • Fréchet spaces: these are complete locally convex spaces where the topology comes from a translation-invariant metric, or equivalently: from a countable family of semi-norms. Many interesting spaces of functions fall into this class. A locally convex F-space is a Fréchet space.
  • Nuclear spaces: a kind of Fréchet space where every bounded map from the nuclear space to an arbitrary Banach space is a nuclear operator.
  • Normed spaces and semi-normed spaces: locally convex spaces where the topology can be described by a single norm or semi-norm. In normed spaces a linear operator is continuous if and only if it is bounded.
  • Banach spaces: Complete normed vector spaces. Most of functional analysis is formulated for Banach spaces.
  • Reflexive Banach spaces: Banach spaces naturally isomorphic to their double dual (see below), which ensures that some geometrical arguments can be carried out. An important example which is not reflexive is L1, whose dual is L but is strictly contained in the dual of L.
  • Hilbert spaces: these have an inner product; even though these spaces may be infinite dimensional, most geometrical reasoning familiar from finite dimensions can be carried out in them.
  • Euclidean spaces: these are finite dimensional Hilbert spaces. According to above comments, any locally compact Hausdorff TVS is isomorphic (as a topological vector space) to one Euclidean space.

In functional analysis, a topological vector space is called locally convex if its topology is defined by a set of convex neighborhoods of 0. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... Look up Convex set in Wiktionary, the free dictionary. ... In functional analysis, given a linear space X, a Minkowski functional is a device that uses the linear structure to introduce a topology on X. // Consider a normed vector space X, with the norm ||·||. Let K be the unit sphere in X. Define a function p : X → R by One... In mathematics, the Hahn-Banach theorem is a central tool in functional analysis. ... In functional analysis and related areas of mathematics barrelled spaces are topological vector spaces where every barrelled set in the space is a neighbourhood for the zero vector. ... In mathematics, the uniform boundedness principle (sometimes known as the Banach-Steinhaus Theorem) is one of the fundamental results of functional analysis. ... In functional analysis and related areas of mathematics a Montel space is a barrelled topological vector space where every closed and bounded set is compact. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In functional analysis and related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be inflated to include the set. ... In mathematics, a compact set is a set of points in a topological space such that every one of its (possibly infinite) open covers has a finite subcover. ... In mathematics, particularly in functional analysis, a bornological space is a locally convex space X such that every semi-norm on X which is bounded on all bounded subsets of X is continuous, where a subset A of X is bounded whenever all continuous semi-norms on X are bounded... In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. ... In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ... In mathematics, an LF-space is a topological vector space V that is a countable strict inductive limit of Fréchet spaces. ... In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts of mathematics, like products and inverse limits. ... In mathematics, the inverse limit (also called the projective limit) is a construction which allows one to glue together several related objects, the precise matter of the gluing process being specified by morphisms between the objects. ... In functional analysis, an F-space is a vector space V over the real or complex numbers together with a metric d : V × V → R so that Scalar multiplication in V is continuous with respect to d and the standard metric on R or C. Addition in V is... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ... This article deals with Fréchet spaces in functional analysis. ... In mathematics a metric or distance is a function which assigns a distance to elements of a set. ... In mathematics the term countable set is used to describe the size of a set, e. ... In mathematics, a nuclear space is a topological vector space with many of the good properties of finite dimensional vector spaces. ... In mathematics, a nuclear operator or a trace-class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ... In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ... In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ... This page concerns the reflexivity of a Banach space. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

Dual space

Every topological vector space has a continuous dual space—the set V* of all continuous linear functionals, i.e. continuous linear maps from the space into the base field K. A topology on the dual can be defined to be the coarsest topology such that the dual pairing V* × V → K is continuous. This turns the dual into a locally convex topological vector space. This topology is called the weak-* topology. This may not be the only natural topology on the dual space; for instance, the dual of a Banach space has a natural norm defined on it. However, it is very important in applications because of its compactness properties (see Banach-Alaoglu theorem). In mathematics it can be shown that any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V. In many cases, these two spaces are isomorphic which means that the distinction between their elements is not always apparent. ... In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. ... In mathematics, weak topology is an alternative term for initial topology. ... The Banach-Alaoglu theorem (also known as Alaoglus theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. ...


References

  • A Grothendieck: Topological vector spaces, Gordon and Breach Science Publishers, New York, 1973. ISBN 0-677-30020-4
  • G Köthe: Topological vector spaces. Grundlehren der mathematischen Wissenschaften, Band 159, Springer-Verlag, New York, 1969. ISBN 0-387-04509-0
  • Schaefer, Helmuth H. (1971). Topological vector spaces. New York: Springer-Verlag. ISBN 0-387-98726-6. 
  • Lang, Serge (1972). Differential manifolds. Reading, Mass.–London–Don Mills, Ont.: Addison-Wesley Publishing Co., Inc.. 
  • F Trèves: Topological Vector Spaces, Distributions, and Kernels, Academic Press, 1967. ISBN 0-486-45352-9.

  Results from FactBites:
 
PlanetMath: topological vector space (152 words)
A topological vector space is necessarily a topological group: the definition ensures that the group operation (vector addition) is continuous, and the inverse operation is the same as multiplication by
A finite-dimensional vector space inherits a natural topology.
This is version 18 of topological vector space, born on 2002-02-03, modified 2007-07-04.
PlanetMath: Fréchet space (436 words)
An F-space is a complete topological vector space whose topology is induced by a translation invariant metric.
Recall that a topological vector space is a uniform space.
A Fréchet space is a complete topological vector space (either real or complex) whose topology is induced by a countable family of semi-norms.
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