# Topological vector space

In

mathematics , a**topological vector space**is one of the basic structures investigated infunctional analysis . As the name suggests the space blends a topological structure (auniform structure to be precise) with thealgebra ic concept of avector space .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.

Hilbert spaces andBanach spaces are well-known examples.**Definition**A

**topological vector space**"X" is avector space over atopological field **K**(most often the real or complex numbers with their standard topologies) which is endowed with a topology such that vector addition "X" × "X" → "X" and scalar multiplication**K**× "X" → "X" are continuous functions.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 ). For a topological vector space to be Hausdorff it suffices that the space be T_{1}.The category of topological vector spaces over a given topological field

**K**is commonly denoted**TVS**_{K}or**TVect**_{K}. The objects are the topological vector spaces over**K**and themorphism s are the continuous**K**-linear maps from one object to another.**Examples**All

normed vector space s, and therefore allBanach space s andHilbert space s, are examples of topological vector spaces.However, there are topological vector spaces whose topology does not arise from a norm, such as spaces of

holomorphic function s on an open domain, spaces ofinfinitely differentiable function s, theSchwartz space s, and spaces oftest function s and the spaces of distributions on them. These are all examples ofMontel space s.A

topological field is a topological vector space over each of itssubfield s.**Product vector spaces**A

cartesian product of a family of topological vector spaces, when endowed with theproduct topology is a topological vector space. For instance, the set "X" of all functions "f" :**R**→**R**. "X" can be identified with the product space**R**^{R}and carries a naturalproduct topology . With this topology, "X" becomes a topological vector space, called the "space ofpointwise convergence ". The reason for this name is the following: if ("f"_{"n"}) is a sequence of elements in "X", then "f"_{"n"}has limit "f" in "X" if and only if "f"_{"n"}("x") has limit "f"("x") for every real number "x". This space is complete, but not normable.**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 abeliantopological group .In particular, topological vector spaces are

uniform space s and one can thus talk about completeness,uniform convergence anduniform 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 adense linear subspace of a complete topological vector space.Vector addition and scalar multiplication are not only continuous but even

homeomorphism s which means we can construct a base for the topology and thus reconstruct the whole topology of the space from anylocal base around the origin.Every topological vector space has a local base of absorbing and

balanced set s.If a topological vector space is

semi-metrisable , that is the topology can be given by asemi-metric , then the semi-metric must be translation invariant. Also, a topological vector space ismetrizable if and only if it is Hausdorff and has a countable local base (i.e., a neighborhood base at the origin).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".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

**K**^{"n"}. A topological vector space isfinite-dimensional if and only if it islocally compact . Here isomorphism means that there exists a linearhomeomorphism between the two spaces.**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 space s: here each point has alocal base consisting ofconvex set s. By a technique known asMinkowski functional s 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 theHahn-Banach theorem .

*Barrelled space s: locally convex spaces where theBanach-Steinhaus theorem holds.

*Montel space : a barrelled space where every closed and bounded set is compact

*Bornological space : a locally convex space where thecontinuous linear operator s to any locally convex space are exactly thebounded linear operator s.

*LF-space s are limits of Fréchet spaces.ILH space s areinverse limit s of Hilbert spaces.

*F-space s are complete topological vector spaces with a translation-invariant metric. These include L^{p}spaces for all p > 0.

*Fréchet space s: these are complete locally convex spaces where the topology comes from atranslation-invariant metric , or equivalently: from acountable family of semi-norms. Many interesting spaces of functions fall into this class. A locally convex F-space is a Fréchet space.

*Nuclear space s: a kind of Fréchet space where every bounded map from the nuclear space to an arbitrary Banach space is anuclear operator .

*Normed space s andsemi-normed space s: 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 space s: Completenormed vector space s. 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 "L"^{1}, whose dual is "L"^{∞}but is strictly contained in the dual of "L"^{∞}.

*Hilbert space s: these have aninner product ; even though these spaces may be infinite dimensional, most geometrical reasoning familiar from finite dimensions can be carried out in them.

*Euclidean space s:**R**^{"n"}or**C**^{"n"}with the topology induced by the standard inner product. As pointed out in the preceding section, for a given finite "n", there is only one topological vector space, up to isomorphism. It follows from this that any finite dimensional subspace of a TVS is closed. A characterization of finite dimensionality is that a Hausdorff TVS is locally compact if and only if it is finite dimensional (therefore isomorphic to some Euclidean space).**Dual space**Every topological vector space has a

continuous dual space —the set "V"^{*}of all continuous linear functionals, i.e.continuous linear map s 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 (seeBanach-Alaoglu theorem ).**References***

*

* cite book

last = Schaefer

first = Helmuth H.

year = 1971

title = Topological vector spaces

series=GTM

volume=3

publisher = Springer-Verlag

location = New York

id = ISBN 0-387-98726-6

* cite book

last=Lang

first=Serge

title=Differential manifolds

publisher=Addison-Wesley Publishing Co., Inc.

location=Reading, Mass.–London–Don Mills, Ont.

year=1972

isbn=0201041669

*

*

*Wikimedia Foundation.
2010.*

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