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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in
functional analysis Functional analysis is a branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), ...
. A topological vector space is a
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
(an algebraic structure) which is also a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
, this implies that vector space operations are
continuous functions In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. More specifically, its topological space has a uniform topological structure, allowing a notion of
uniform convergenceIn the mathematical field of analysis, uniform convergence is a mode Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Language * Grammatical mode or grammatical mood, a category of verbal inflections t ...
. The elements of topological vector spaces are typically
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
s or
linear operators In mathematics, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \rightarrow W between two vector spaces that preserves the operat ...

linear operators
acting on topological vector spaces, and the topology is often defined so as to capture a particular notion of
convergence Convergence may refer to: Arts and media Literature *Convergence (book series), ''Convergence'' (book series), edited by Ruth Nanda Anshen *Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics: **A four-par ...
of sequences of functions.
Banach space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s,
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s and
Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a normed space, norm that is a combination of Lp norm, ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a s ...
s are well-known examples. Unless stated otherwise, the underlying field of a topological vector space is assumed to be either the
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s \Complex or the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s \R.


Motivation

Normed spaces Every
normed vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
has a natural
topological structure In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
: the norm induces a
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
and the metric induces a topology. This is a topological vector space because: #The vector addition \cdot\, + \,\cdot\; : X \times X \to X is jointly continuous with respect to this topology. This follows directly from the
triangle inequality In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

triangle inequality
obeyed by the norm. #The scalar multiplication \cdot : \mathbb \times X \to X, where \mathbb is the underlying scalar field of X, is jointly continuous. This follows from the triangle inequality and homogeneity of the norm. Thus all
Banach space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s and
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s are examples of topological vector spaces. Non-normed spaces There are topological vector spaces whose topology is not induced by a norm, but are still of interest in analysis. Examples of such spaces are spaces of
holomorphic function In mathematics, a holomorphic function is a complex-valued function Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), ...
s on an open domain, spaces of
infinitely differentiable function In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mat ...
s, the
Schwartz space Schwartz may refer to: *Schwartz (surname), a surname (and list of people with the name) *Schwartz (brand), a spice brand *Schwartz's, a delicatessen in Montreal, Quebec, Canada *Schwartz Publishing, an Australian publishing house *"Danny Schwartz", ...
s, and spaces of
test function Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to derivative, differentiate functions whose deri ...
s and the spaces of distributions on them. These are all examples of
Montel space In functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional a ...
s. An infinite-dimensional Montel space is never normable. The existence of a norm for a given topological vector space is characterized by
Kolmogorov's normability criterionIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. A
topological fieldIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
is a topological vector space over each of its subfields.


Definition

A topological vector space (TVS) X is a
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
over a
topological fieldIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
\mathbb (most often the
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

complex
numbers with their standard topologies) that is endowed with a
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
such that vector addition \cdot\, + \,\cdot\; : X \times X \to X and scalar multiplication \cdot : \mathbb \times X \to X are
continuous functions In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(where the domains of these functions are endowed with product topologies). Such a topology is called a or a on X. Every topological vector space is also a commutative
topological group In mathematics, topological groups are logically the combination of Group (mathematics), groups and Topological space, topological spaces, i.e. they are groups and topological spaces at the same time, such that the Continuous function, continui ...
under addition. Hausdorff assumption Some authors (for example,
Walter Rudin Walter Rudin (May 2, 1921 – May 20, 2010) was an Austria, Austrian-United States, American mathematician and professor of Mathematics at the University of Wisconsin–Madison. In addition to his contributions to complex and harmonic analysis, R ...
) require the topology on X to be T1; it then follows that the space is
Hausdorff
Hausdorff
, and even
TychonoffTikhonov (russian: Ти́хонов, link=no; masculine), sometimes spelled as Tychonoff, or Tikhonova (; feminine) is a Russian language, Russian surname that is derived from the male given name Tikhon, the Russian form of the Greek name Τύχων ...
. A topological vector space is said to be if it is Hausdorff; importantly, "separated" does not mean separable. The topological and linear algebraic structures can be tied together even more closely with additional assumptions, the most common of which are listed
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Fred Below (1926–1988), American blues drummer *Fritz von Below (1853 ...
. Category and morphisms The
category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
of topological vector spaces over a given topological field \mathbb is commonly denoted TVS\mathbb or TVect\mathbb. The objects are the topological vector spaces over \mathbb and the
morphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

morphism
s are the continuous \mathbb-linear maps from one object to another. A (abbreviated ) or
topological homomorphism In functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis ...
is a
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
linear map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

linear map
u : X \to Y between topological vector spaces (TVSs) such that the induced map u : X \to \operatorname u is an
open mapping In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
when \operatorname u := u(X), which is the range or image of u, is given the
subspace topologyIn topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a Topological_space#Definitions, topology induced from that of ''X'' called the subspace topology (or the relative ...

subspace topology
induced by ''Y''. A (abbreviated ) or a topological
monomorphism In the context of abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, rin ...
is an
injective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
topological homomorphism. Equivalently, a TVS-embedding is a linear map that is also a
topological embedding In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. A (abbreviated ), also called a or an , is a bijective
linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out se ...

linear
homeomorphism In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantiti ...
. Equivalently, it is a
surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
TVS embedding Many properties of TVSs that are studied, such as local convexity, metrizability, completeness, and normability, are invariant under TVS isomorphisms. A necessary condition for a vector topology A collection \mathcal of subsets of a vector space is called additive if for every N \in \mathcal, there exists some U \in \mathcal such that U + U \subseteq N. All of the above conditions are consequently a necessity for a topology to form a vector topology.


Defining topologies using neighborhoods of the origin

Since every vector topology is translation invariant (which means that for all x_0 \in X, the map X \to X defined by x \mapsto x_0 + x is a
homeomorphism In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantiti ...
), to define a vector topology it suffices to define a
neighborhood basisIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
(or subbasis) for it at the origin. In general, the set of all balanced and absorbing subsets of a vector space does not satisfy the conditions of this theorem and does not form a neighborhood basis at the origin for any vector topology.


Defining topologies using strings

Let X be a vector space and let U_ = \left(U_i\right)_^ be a sequence of subsets of X. Each set in the sequence U_ is called a of U_ and for every index i, U_i is called the ith knot of U_. The set U_1 is called the beginning of U_. The sequence U_ is/is a: * if U_ + U_ \subseteq U_i for every index i. *
Balanced In telecommunication Telecommunication is the transmission of information Information can be thought of as the resolution of uncertainty; it answers the question of "What an entity is" and thus defines both its essence and the nature of i ...
(resp. absorbing, closed,The topological properties of course also require that X be a TVS. convex, open,
symmetric Symmetry (from Greek συμμετρία ''symmetria'' "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more pre ...
, barrelled, absolutely convex/disked, etc.) if this is true of every U_i. * if U_ is summative, absorbing, and balanced. * or a in a TVS X if U_ is a string and each of its knots is a neighborhood of the origin in X. If Uis an absorbing
disk Disk or disc may refer to: * Disk (mathematics) * Disk storage Music * Disc (band), an American experimental music band * Disk (album), ''Disk'' (album), a 1995 EP by Moby Other uses * Disc (galaxy), a disc-shaped group of stars * Disc (magazin ...
in a vector space X then the sequence defined by U_i := 2^ U forms a string beginning with U_1 = U. This is called the natural string of U Moreover, if a vector space X has countable dimension then every string contains an
absolutely convexIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
string. Summative sequences of sets have the particularly nice property that they define non-negative continuous real-valued
subadditiveIn mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two element (set), elements of the Domain of a function, domain always returns something less than or equal to the sum of the ...
functions. These functions can then be used to prove many of the basic properties of topological vector spaces. A proof of the above theorem is given in the article on metrizable TVSs. If U_ = \left(U_i\right)_ and V_ = \left(V_i\right)_ are two collections of subsets of a vector space X and if s is a scalar, then by definition: * V_ contains U_: \ U_ \subseteq V_ if and only if U_i \subseteq V_i for every index i. * Set of knots: \ \operatorname U_ := \left\. * Kernel: \ \ker U_ := \bigcap_ U_i. * Scalar multiple: \ s U_ := \left(s U_i\right)_. * Sum: \ U_ + V_ := \left(U_i + V_i\right)_. * Intersection: \ U_ \cap V_ := \left(U_i \cap V_i\right)_.. If \mathbb is a collection sequences of subsets of X, then \mathbb is said to be directed (downwards) under inclusion or simply directed if \mathbb is not empty and for all U_, V_ \in \mathbb, there exists some W_ \in \mathbb such that W_ \subseteq U_ and W_ \subseteq V_ (said differently, if and only if \mathbb is a prefilter with respect to the containment \,\subseteq\, defined above). Notation: Let \operatorname \mathbb := \bigcup_ \operatorname U_. be the set of all knots of all strings in \mathbb. Defining vector topologies using collections of strings is particularly useful for defining classes of TVSs that are not necessarily locally convex. If \mathbb is the set of all topological strings in a TVS (X, \tau) then \tau_ = \tau. A Hausdorff TVS is
metrizable In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...
if and only if its topology can be induced by a single topological string.


Topological structure

A vector space is an
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
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, topological groups are logically the combination of Group (mathematics), groups and Topological space, topological spaces, i.e. they are groups and topological spaces at the same time, such that the Continuous function, continui ...
. Every TVS is
completely regular In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. Tychonoff spaces are named after Andrey Nikolayevich Tychonoff, w ...
but a TVS need not be
normal
normal
. Let X be a topological vector space. Given a
subspace
subspace
M \subseteq X, the quotient space X / M with the usual
quotient topology In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...
is a Hausdorff topological vector space if and only if M is closed.In particular, X is Hausdorff if and only if the set \ is closed (that is, X is a T1 space). This permits the following construction: given a topological vector space X (that is probably not Hausdorff), form the quotient space X / M where M is the closure of \. X / M is then a Hausdorff topological vector space that can be studied instead of X.


Invariance of vector topologies

One of the most used properties of vector topologies is that every vector topology is translation invariant: :for all x_0 \in X, the map X \to X defined by x \mapsto x_0 + x is a
homeomorphism In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantiti ...
, but if x_0 \neq 0 then it is not linear and so not a TVS-isomorphism. Scalar multiplication by a non-zero scalar is a TVS-isomorphism. This means that if s \neq 0 then the linear map X \to X defined by x \mapsto s x is a homeomorphism. Using s = -1 produces the negation map X \to X defined by x \mapsto - x, which is consequently a linear homeomorphism and thus a TVS-isomorphism. If x \in X and any subset S \subseteq X, then \operatorname_X (x + S) = x + \operatorname_X S and moreover, if 0 \in S then x + S is a
neighborhood A neighbourhood (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...
(resp. open neighborhood, closed neighborhood) of x in X if and only if the same is true of S at the origin.


Local notions

A subset E of a vector space X is said to be * absorbing (in X): if for every x \in X, there exists a real r > 0 such that c x \in E for any scalar c satisfying , c, \leq r. *
balanced In telecommunication Telecommunication is the transmission of information Information can be thought of as the resolution of uncertainty; it answers the question of "What an entity is" and thus defines both its essence and the nature of i ...
or circled: if t E \subseteq E for every scalar , t, \leq 1. *
convex Convex means curving outwards like a sphere, and is the opposite of concave. Convex or convexity may refer to: Science and technology * Convex lens A lens is a transmissive optics, optical device which focuses or disperses a light beam by me ...

convex
: if t E + (1 - t) E \subseteq E for every real 0 \leq t \leq 1. * a
disk Disk or disc may refer to: * Disk (mathematics) * Disk storage Music * Disc (band), an American experimental music band * Disk (album), ''Disk'' (album), a 1995 EP by Moby Other uses * Disc (galaxy), a disc-shaped group of stars * Disc (magazin ...
or
absolutely convexIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
: if E is convex and balanced. *
symmetric Symmetry (from Greek συμμετρία ''symmetria'' "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more pre ...
: if - E \subseteq E, or equivalently, if - E = E. Every neighborhood of 0 is an
absorbing set In functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis ...
and contains an open
balanced In telecommunication Telecommunication is the transmission of information Information can be thought of as the resolution of uncertainty; it answers the question of "What an entity is" and thus defines both its essence and the nature of i ...
neighborhood of 0 so every topological vector space has a local base of absorbing and
balanced set In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field (mathematics), field with an absolute value (algebra), absolute value function , \cdot , ) is a Set (mathematics), set such tha ...
s. The origin even has a neighborhood basis consisting of closed balanced neighborhoods of 0; if the space is
locally convex 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 spa ...
then it also has a neighborhood basis consisting of closed convex balanced neighborhoods of 0. Bounded subsets A subset E of a topological vector space X is bounded if for every neighborhood V of the origin, then E \subseteq t V when t is sufficiently large. The definition of boundedness can be weakened a bit; E is bounded if and only if every countable subset of it is bounded. A set is bounded if and only if each of its subsequences is a bounded set. Also, E is bounded if and only if for every balanced neighborhood V of 0, there exists t such that E \subseteq t V. Moreover, when X is locally convex, the boundedness can be characterized by
seminorm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...
s: the subset E is bounded if and only if every continuous seminorm p is bounded on E. Every
totally boundedIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structu ...
set is bounded. If M is a vector subspace of a TVS X, then a subset of M is bounded in M if and only if it is bounded in X.


Metrizability

A TVS is pseudometrizable if and only if it has a countable neighborhood basis at the origin, or equivalent, if and only if its topology is generated by an ''F''-seminorm. A TVS is metrizable if and only if it is Hausdorff and pseudometrizable. More strongly: a topological vector space is said to be
normable In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
if its topology can be induced by a norm. A topological vector space is normable if and only if it is Hausdorff and has a convex bounded neighborhood of 0. Let \mathbb be a non-
discrete Discrete in science is the opposite of :wikt:continuous, continuous: something that is separate; distinct; individual. Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic c ...
locally compact In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
topological field, for example the real or complex numbers. A
Hausdorff
Hausdorff
topological vector space over \mathbb is locally compact if and only if it is
finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e. the number of vectors) of a Basis (linear algebra), basis of ''V'' over its base Field (mathematics), field. p. 44, §2.36 It is sometimes called Hamel dimension (after ...
, that is, isomorphic to \mathbb^n for some natural number n.


Completeness and uniform structure

The canonical uniformity on a TVS (X, \tau) is the unique translation-invariant
uniformity Uniformity may refer to: * Distribution uniformity, a measure of how uniformly water is applied to the area being watered * Religious uniformity, the promotion of one state religion, denomination, or philosophy to the exclusion of all other religi ...
that induces the topology \tau on X. Every TVS is assumed to be endowed with this canonical uniformity, which makes all TVSs into
uniform space In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s. This allows one to about related notions such as completeness,
uniform convergenceIn the mathematical field of analysis, uniform convergence is a mode Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Language * Grammatical mode or grammatical mood, a category of verbal inflections t ...
, Cauchy nets, and
uniform continuity In mathematics, a function (mathematics), function ''f'' is uniformly continuous if, roughly speaking, it is possible to guarantee that ''f''(''x'') and ''f''(''y'') be as close to each other as we please by requiring only that ''x'' and ''y'' be s ...
. etc., which are always assumed to be with respect to this uniformity (unless indicated other). This implies that every Hausdorff topological vector space is
TychonoffTikhonov (russian: Ти́хонов, link=no; masculine), sometimes spelled as Tychonoff, or Tikhonova (; feminine) is a Russian language, Russian surname that is derived from the male given name Tikhon, the Russian form of the Greek name Τύχων ...
. A subspace of a TVS is
compact Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
if and only if it is complete and
totally boundedIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structu ...
(for Hausdorff TVSs, a set being totally bounded is equivalent to it being precompact). But if the TVS is not Hausdorff then there exist compact subsets that are not closed. However, the closure of a compact subset of a non-Hausdorff TVS is again compact (so compact subsets are
relatively compact In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
). With respect to this uniformity, a
net Net or net may refer to: Mathematics and physics * Net (mathematics) In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they ...
(or sequence) x_ = \left(x_i\right)_ is Cauchy if and only if for every neighborhood V of 0, there exists some index i such that x_m - x_n \in V whenever j \geq i and k \geq i. Every Cauchy sequence is bounded, although Cauchy nets and Cauchy filters may not be bounded. A topological vector space where every Cauchy sequence converges is called sequentially complete; in general, it may not be complete (in the sense that all Cauchy filters converge). The vector space operation of addition is uniformly continuous and an
open map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. Scalar multiplication is Cauchy continuous but in general, it is almost never uniformly continuous. Because of this, every topological vector space can be completed and is thus a Dense set, dense linear subspace of a complete topological vector space. * Every TVS has a Complete topological vector space, completion and every Hausdorff TVS has a Hausdorff completion. Every TVS (even those that are Hausdorff and/or complete) has infinitely many non-isomorphic non-Hausdorff completions. * A compact subset of a TVS (not necessarily Hausdorff) is complete. A complete subset of a Hausdorff TVS is closed. * If C is a complete subset of a TVS then any subset of C that is closed in C is complete. * A Cauchy sequence in a Hausdorff TVS X is not necessarily
relatively compact In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(that is, its closure in X is not necessarily compact). * If a Cauchy filter in a TVS has an Filters in topology, accumulation point x then it converges to x. * If a series \sum_^ x_i convergesA series \sum_^ x_i is said to converge in a TVS X if the sequence of partial sums converges. in a TVS X then x_ \to 0 in X.


Examples


Finest and coarsest vector topology

Let X be a real or complex vector space. Trivial topology The trivial topology or indiscrete topology \ is always a TVS topology on any vector space X and it is the coarsest TVS topology possible. An important consequence of this is that the intersection of any collection of TVS topologies on X always contains a TVS topology. Any vector space (including those that are infinite dimensional) endowed with the trivial topology is a compact (and thus
locally compact In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
) Complete topological vector space, complete Metrizable topological vector space, pseudometrizable Seminormed space, seminormable Locally convex topological vector space, locally convex topological vector space. It is
Hausdorff
Hausdorff
if and only if \operatorname X = 0. Finest vector topology There exists a TVS topology \tau_f on X, called the on X, that is finer than every other TVS-topology on X (that is, any TVS-topology on X is necessarily a subset of \tau_f). Every linear map from \left(X, \tau_f\right) into another TVS is necessarily continuous. If X has an uncountable Hamel basis then \tau_f is Locally convex topological vector space, locally convex and
metrizable In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...
.


Product vector spaces

A Cartesian product of a family of topological vector spaces, when endowed with the product topology, is a topological vector space. Consider for instance the set X of all functions f: \R \to \R where \R carries its usual Euclidean topology. This set X is a real vector space (where addition and scalar multiplication are defined pointwise, as usual) that can be identified with (and indeed, is often defined to be) the Cartesian product \R^\R,, which carries the natural product topology. With this product topology, X := \R^ becomes a topological vector space whose topology is called . The reason for this name is the following: if \left(f_n\right)_^ is a sequence (or more generally, a
net Net or net may refer to: Mathematics and physics * Net (mathematics) In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they ...
) of elements in X and if f \in X then f_n limit of a sequence, converges to f in X if and only if for every real number x, f_n(x) converges to f(x) in \R. This TVS is Complete topological vector space, complete,
Hausdorff
Hausdorff
, and
locally convex 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 spa ...
but not
metrizable In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...
and consequently not
normable In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
; indeed, every neighborhood of the origin in the product topology contains lines (that is, 1-dimensional vector subspaces, which are subsets of the form \R f := \ with f \neq 0).


Finite-dimensional spaces

By F. Riesz's theorem, a Hausdorff topological vector space is finite-dimensional if and only if it is
locally compact In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
, which happens if and only if it has a compact
neighborhood A neighbourhood (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...
of the origin. Let \mathbb denote \R or \Complex and endow \mathbb with its usual Hausdorff normed Euclidean topology. Let X be a vector space over \mathbb of finite dimension n := \operatorname X and so that X is vector space isomorphic to \mathbb^n (explicitly, this means that there exists a linear isomorphism between the vector spaces X and \mathbb^n). This finite-dimensional vector space X always has a unique vector topology, which makes it TVS-isomorphic to \mathbb^n, where \mathbb^n is endowed with the usual Euclidean topology (which is the same as the product topology). This Hausdorff vector topology is also the (unique) Comparison of topologies, finest vector topology on X. X has a unique vector topology if and only if \operatorname X = 0. If \operatorname X \neq 0 then although X does not have a unique vector topology, it does have a unique vector topology. * If \operatorname X = 0 then X = \ has exactly one vector topology: the trivial topology, which in this case (and in this case) is Hausdorff. The trivial topology on a vector space is Hausdorff if and only if the vector space has dimension 0. * If \operatorname X = 1 then X has two vector topologies: the usual Euclidean topology and the (non-Hausdorff) trivial topology. ** Since the field \mathbb is itself a 1-dimensional topological vector space over \mathbb and since it plays an important role in the definition of topological vector spaces, this dichotomy plays an important role in the definition of an
absorbing set In functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis ...
and has consequences that reverberate throughout
functional analysis Functional analysis is a branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), ...
. The proof of this dichotomy is straightforward so only an outline with the important observations is given. As usual, \mathbb is assumed have the (normed) Euclidean topology. Let B_r := \ for all r > 0. Let X be a 1-dimensional vector space over \mathbb. If S \subseteq X and B \subseteq \mathbb is a ball centered at 0 then B \cdot S = X whenever S contains an "unbounded sequence", by which it is meant a sequence of the form \left(a_i x\right)_^ where 0 \neq x \in X and \left(a_i\right)_^ \subseteq \mathbb is unbounded in normed space \mathbb (in the usual sense). Any vector topology on X will be translation invariant and invariant under non-zero scalar multiplication, and for every 0 \neq x \in X, the map M_x : \mathbb \to X given by M_x(a) := a x is a continuous linear bijection. Because X = \mathbb x for any such x, every subset of X can be written as F x = M_x(F) for some unique subset F \subseteq \mathbb. And if this vector topology on X has a neighborhood W of the origin that is not equal to all of X, then the continuity of scalar multiplication \mathbb \times X \to X at the origin guarantees the existence of an open ball B_r \subseteq \mathbb centered at 0 and an open neighborhood S of the origin in X such that B_r \cdot S \subseteq W \neq X, which implies that S does contain any "unbounded sequence". This implies that for every 0 \neq x \in X, there exists some positive integer n such that S \subseteq B_n x. From this, it can be deduced that if X does not carry the trivial topology and if 0 \neq x \in X, then for any ball B \subseteq \mathbb center at 0 in \mathbb, M_x(B) = B x contains an open neighborhood of the origin in X, which then proves that M_x is a linear
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. \blacksquare * If \operatorname X = n \geq 2 then X has distinct vector topologies: ** Some of these topologies are now described: Every linear functional f on X, which is vector space isomorphic to \mathbb^n,, induces a
seminorm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...
, f, : X \to \R defined by , f, (x) = , f(x), where \ker f = \ker , f, . Every seminorm induces a ( pseudometrizable
locally convex 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 spa ...
) vector topology on X and seminorms with distinct kernels induce distinct topologies so that in particular, seminorms on X that are induced by linear functionals with distinct kernel will induces distinct vector topologies on X. ** However, while there are infinitely many vector topologies on X when \operatorname X \geq 2, there are, only 1 + \operatorname X vector topologies on X. For instance, if n := \operatorname X = 2 then the vector topologies on X consist of the trivial topology, the Hausdorff Euclidean topology, and then the infinitely many remaining non-trivial non-Euclidean vector topologies on X are all TVS-isomorphic to one another.


Non-vector topologies

Discrete and cofinite topologies If X is a non-trivial vector space (that is, of non-zero dimension) then the discrete topology on X (which is always Metrizable space, metrizable) is a TVS topology because despite making addition and negation continuous (which makes it into a
topological group In mathematics, topological groups are logically the combination of Group (mathematics), groups and Topological space, topological spaces, i.e. they are groups and topological spaces at the same time, such that the Continuous function, continui ...
under addition), it fails to make scalar multiplication continuous. The cofinite topology on X (where a subset is open if and only if its complement is finite) is also a TVS topology on X.


Linear maps

A linear operator between two topological vector spaces which is continuous at one point is continuous on the whole domain. Moreover, a linear operator f is continuous if f(X) is bounded (as defined below) for some neighborhood X of the origin. A hyperplane on a topological vector space X is either dense or closed. A linear functional f on a topological vector space X has either dense or closed kernel. Moreover, f is continuous if and only if its Kernel (algebra), kernel is closed set, closed.


Types

Depending on the application additional constraints are usually enforced on the topological structure of the space. In fact, several principal results in functional analysis fail to hold in general for topological vector spaces: the closed graph theorem, the Open mapping theorem (functional analysis), open mapping theorem, and the fact that the dual space of the space separates points in the space. Below are some common topological vector spaces, roughly in order of increasing "niceness." * F-spaces are complete space, complete topological vector spaces with a translation-invariant metric. These include Lp space, L^p spaces for all p > 0. * 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 seminorms. Local convexity is the minimum requirement for "geometrical" arguments like the Hahn–Banach theorem. The L^p spaces are locally convex (in fact, Banach spaces) for all p \geq 1, but not for 0 < p < 1. * Barrelled spaces: locally convex spaces where the Banach–Steinhaus theorem holds. * Bornological space: a locally convex space where the continuous linear operators to any locally convex space are exactly the bounded linear operators. * Stereotype space: a locally convex space satisfying a variant of reflexive space, reflexivity condition, where the dual space is endowed with the topology of uniform convergence on totally bounded space, totally bounded sets. *
Montel space In functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional a ...
: a barrelled space where every closed set, closed and Bounded set (topological vector space), bounded set is compact set, compact * Fréchet spaces: these are complete locally convex spaces where the topology comes from a translation-invariant metric, or equivalently: from a countable family of seminorms. Many interesting spaces of functions fall into this class -- C^\infty(\R) is a Fréchet space under the seminorms \, f\, _ = \sup_ , f^(x), . A locally convex F-space is a Fréchet space. * LF-spaces are limit (category theory), limits of Fréchet spaces. ILH spaces are inverse limits of Hilbert spaces. * Nuclear spaces: these are locally convex spaces with the property that every bounded map from the nuclear space to an arbitrary Banach space is a nuclear operator. * Normed spaces and seminormed spaces: locally convex spaces where the topology can be described by a single norm (mathematics), norm or seminorm (mathematics), seminorm. In normed spaces a linear operator is continuous if and only if it is bounded. *
Banach space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s: Complete
normed vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s. Most of functional analysis is formulated for Banach spaces. This class includes the L^p spaces with 1\leq p \leq \infty, the space BV of Bounded variation, functions of bounded variation, and Ba space, certain spaces of measures. * Reflexive space, 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 reflexive is Lp space, L^1, whose dual is L^ but is strictly contained in the dual of L^. *
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s: 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. These include L^2 spaces, the L^2 Sobolev space, Sobolev spaces W^, and Hardy space, Hardy spaces. * Euclidean spaces: \R^n or \Complex^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 n-dimensional 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 X^ of all continuous linear functionals, that is, continuous linear maps from the space into the base field \mathbb. A topology on the dual can be defined to be the coarsest topology such that the dual pairing each point evaluation X^ \to \mathbb is continuous. This turns the dual into a locally convex topological vector space. This topology is called the Weak topology, weak-* topology. This may not be the only natural topology on the dual space; for instance, the dual of a normed space has a natural norm defined on it. However, it is very important in applications because of its compactness properties (see Banach–Alaoglu theorem). Caution: Whenever X is a non-normable locally convex space, then the pairing map X^ \times X \to \mathbb is never continuous, no matter which vector space topology one chooses on X^.


Properties

For any S \subseteq X of a TVS X, the Convex set, convex (resp.
balanced In telecommunication Telecommunication is the transmission of information Information can be thought of as the resolution of uncertainty; it answers the question of "What an entity is" and thus defines both its essence and the nature of i ...
, Absolutely convex set, disked, closed convex, closed balanced, closed disked) hull of S is the smallest subset of X that has this property and contains S. The closure (resp. interior, convex hull, balanced hull, disked hull) of a set S is sometimes denoted by \operatorname_X S (resp. \operatorname_X S, \operatorname S, \operatorname S, \operatorname S).


Neighborhoods and open sets

Properties of neighborhoods and open sets Every TVS is Connected space, connected and Locally connected space, locally connected and any connected open subset of a TVS is arcwise connected. If S \subseteq X and U is an open subset of X then S + U is an open set in X and if S \subseteq X has non-empty interior then S - S is a neighborhood of the origin. The open convex subsets of a TVS X (not necessarily Hausdorff or locally convex) are exactly those that are of the form z + \ ~=~ \ for some z \in X and some positive continuous sublinear functional p on X. If K is an absorbing
disk Disk or disc may refer to: * Disk (mathematics) * Disk storage Music * Disc (band), an American experimental music band * Disk (album), ''Disk'' (album), a 1995 EP by Moby Other uses * Disc (galaxy), a disc-shaped group of stars * Disc (magazin ...
in a TVS X and if p := p_K is the Minkowski functional of K then \operatorname_X K ~\subseteq~ \ ~\subseteq~ K ~\subseteq~ \ ~\subseteq~ \operatorname_X K where importantly, it was assumed that K had any topological properties nor that p was continuous (which happens if and only if K is a neighborhood of 0). Let \tau and \nu be two vector topologies on X. Then \tau \subseteq \nu if and only if whenever a net x_ = \left(x_i\right)_ in X converges 0 in (X, \nu) then x_ \to 0 in (X, \tau). Let \mathcal be a neighborhood basis of the origin in X, let S \subseteq X, and let x \in X. Then x \in \operatorname_X S if and only if there exists a net s_ = \left(s_N\right)_ in S (indexed by \mathcal) such that s_ \to x in X. This shows, in particular, that it will often suffice to consider nets indexed by a neighborhood basis of the origin rather than nets on arbitrary directed sets. Interior If R, S \subseteq X and S has non-empty interior then \operatorname_X S ~=~ \operatorname_X \left(\operatorname_X S\right)~ \text ~\operatorname_X S ~=~ \operatorname_X \left(\operatorname_X S\right) and \operatorname_X (R) + \operatorname_X (S) ~\subseteq~ R + \operatorname_X S \subseteq \operatorname_X (R + S). If S is a
disk Disk or disc may refer to: * Disk (mathematics) * Disk storage Music * Disc (band), an American experimental music band * Disk (album), ''Disk'' (album), a 1995 EP by Moby Other uses * Disc (galaxy), a disc-shaped group of stars * Disc (magazin ...
in X that has non-empty interior then the origin belongs to the interior of S. However, a closed
balanced In telecommunication Telecommunication is the transmission of information Information can be thought of as the resolution of uncertainty; it answers the question of "What an entity is" and thus defines both its essence and the nature of i ...
subset of X with non-empty interior may fail to contain the origin in its interior. If S is a
balanced In telecommunication Telecommunication is the transmission of information Information can be thought of as the resolution of uncertainty; it answers the question of "What an entity is" and thus defines both its essence and the nature of i ...
subset of X with non-empty interior then \ \cup \operatorname_X S is balanced; in particular, if the interior of a balanced set contains the origin then \operatorname_X S is balanced.If the interior of a balanced set is non-empty but does not contain the origin (such sets exists even in \R^2 and \Complex^2) then the interior of this set can not be a balanced set. If C is convex and 0 < t \leq 1, then t \operatorname C + (1 - t) \operatorname C ~\subseteq~ \operatorname C. If x belongs to the interior of a convex set S \subseteq X and y \in \operatorname_X S, then the half-open line segment [x, y) := \ \subseteq \operatorname_X \text x \neq y and [x, x) = \varnothing \text x = y. If N is a
balanced In telecommunication Telecommunication is the transmission of information Information can be thought of as the resolution of uncertainty; it answers the question of "What an entity is" and thus defines both its essence and the nature of i ...
neighborhood of 0 in X and B_1 := \, then by considering intersections of the form N \cap \R x (which are convex
symmetric Symmetry (from Greek συμμετρία ''symmetria'' "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more pre ...
neighborhoods of 0 in the real TVS \R x) it follows that: \operatorname N = [0, 1) \operatorname N = (-1, 1) N = B_1 N, and furthermore, if x \in \operatorname N \text r := \sup_ \ then r > 1 \text [0, r) x \subseteq \operatorname N, and if r \neq \infty then r x \in \operatorname N \setminus \operatorname N.


Non-Hausdorff spaces and the closure of the origin

A topological vector space X is Hausdorff if and only if \ is a closed subset of X, or equivalently, if and only if \ = \operatorname_X \. Because \ is a vector subspace of X, the same is true of its closure \operatorname_X \, which is referred to as in X. This vector space satisfies \operatorname_X \ = \bigcap_ N so that in particular, every neighborhood of the origin in X contains the vector space \operatorname_X \ as a subset. The
subspace topologyIn topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a Topological_space#Definitions, topology induced from that of ''X'' called the subspace topology (or the relative ...

subspace topology
on \operatorname_X \ is always the trivial topology, which in particular implies that the topological vector space \operatorname_X \ a compact space (even if its dimension is non-zero or even infinite) and consequently also a Bounded set (topological vector space), bounded subset of X. In fact, a vector subspace of a TVS is bounded if and only if it is contained in the closure of \. Every subset of \operatorname_X \ also carries the trivial topology and so is itself a compact, and thus also complete, Topological subspace, subspace (see footnote for a proof).Since \operatorname_X \ has the trivial topology, so does each of its subsets, which makes them all compact. It is known that a subset of any uniform space is compact if and only if it is complete and totally bounded. In particular, if X is not Hausdorff then there exist subsets that are both but in X; for instance, this will be true of any non-empty proper subset of \operatorname_X \. If S \subseteq X is compact, then \operatorname_X S = S + \operatorname_X \ and this set is compact. Thus the closure of a compact subset of a TVS is compact (said differently, all compact sets are
relatively compact In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
), which is not guaranteed for arbitrary non-Hausdorff
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
s.In general topology, the closure of a compact subset of a non-Hausdorff space may fail to be compact (for example, the particular point topology on an infinite set). This result shows that this does not happen in non-Hausdorff TVSs. S + \operatorname_X \ is compact because it is the image of the compact set S \times \operatorname_X \ under the continuous addition map \cdot\, + \,\cdot\; : X \times X \to X. Recall also that the sum of a compact set (i.e. S) and a closed set is closed so S + \operatorname_X \ is closed in X. For every subset S \subseteq X, S + \operatorname_X \ \subseteq \operatorname_X S and consequently, if S \subseteq X is open or closed in X then S + \operatorname_X \ = SIf s \in S then s + \operatorname_X \ = \operatorname_X (s + \) = \operatorname_X \ \subseteq \operatorname_X S. Because S \subseteq S + \operatorname_X \ \subseteq \operatorname_X S, if S is closed then equality holds. Using the fact that \operatorname_X \ is a vector space, it is readily verified that the complement in X of any set S satisfying the equality S + \operatorname_X \ = S must also satisfy this equality (when X \setminus S is substituted for S). (so that this open closed subsets S can be described as a Tube lemma, "tube" whose vertical side is the vector space \operatorname_X \). For any subset S \subseteq X of this TVS X, the following are equivalent: * S is Totally bounded space, totally bounded. * S + \operatorname_X \ is totally bounded. * \operatorname_X S is totally bounded. * The image if S under the canonical quotient map X \to X / \operatorname_X (\) is totally bounded. If M is a vector subspace of a TVS X then X / M is Hausdorff if and only if M is closed in X. Moreover, the quotient map q : X \to X / \operatorname_X \ is always a Open and closed maps, closed map onto the (necessarily) Hausdorff TVS. Every vector subspace of X that is an algebraic complement of \operatorname_X \ (that is, a vector subspace H that satisfies \ = H \cap \operatorname_X \ and X = H + \operatorname_X \) is a Complemented subspace, topological complement of \operatorname_X \. Consequently, if H is an algebraic complement of \operatorname_X \ in X then the addition map H \times \operatorname_X \ \to X, defined by (h, n) \mapsto h + n is a TVS-isomorphism, where H is necessarily Hausdorff and \operatorname_X \ has the indiscrete topology. Moreover, if C is a Hausdorff Complete topological vector space, completion of H then C \times \operatorname_X \ is a completion of X \cong H \times \operatorname_X \.


Closed and compact sets

Compact and totally bounded sets A subset of a TVS is compact if and only if it is complete and Totally bounded space, totally bounded. Thus, in a complete topological vector space, a closed and totally bounded subset is compact. A subset S of a TVS X is Totally bounded space, totally bounded if and only if \operatorname_X S is totally bounded, if and only if its image under the canonical quotient map X \to X / \operatorname_X (\) is totally bounded. Every relatively compact set is totally bounded and the closure of a totally bounded set is totally bounded. The image of a totally bounded set under a uniformly continuous map (such as a continuous linear map for instance) is totally bounded. If S is a subset of a TVS X such that every sequence in S has a cluster point in S then S is totally bounded. If K is a compact subset of a TVS X and U is an open subset of X containing K, then there exists a neighborhood N of 0 such that K + N \subseteq U. Closure and closed set The closure of any convex (respectively, any balanced, any absorbing) subset of any TVS has this same property. In particular, the closure of any convex, balanced, and absorbing subset is a Barrelled space#barrel, barrel. The closure of a vector subspace of a TVS is a vector subspace. Every finite dimensional vector subspace of a Hausdorff TVS is closed. The sum of a closed vector subspace and a finite-dimensional vector subspace is closed. If M is a vector subspace of X and N is a closed neighborhood of the origin in X such that U \cap N is closed in X then M is closed in X. The sum of a compact set and a closed set is closed. However, the sum of two closed subsets may fail to be closed (see this footnoteIn the \R^2,, the sum of the y-axis and the graph of y = \frac, which is the complement of the y-axis, is open in \R^2. In \R, the Minkowski sum \Z + \sqrt\Z is a countable dense subset of \R so not closed in \R. for examples). If S \subseteq X and a is a scalar then a \operatorname_X S \subseteq \operatorname_X (a S), where if X is Hausdorff, a \neq 0, \text S = \varnothing then equality holds: \operatorname_X (a S) = a \operatorname_X S. In particular, every non-zero scalar multiple of a closed set is closed. If S \subseteq X and if A is a set of scalars such that neither \operatorname S \text \operatorname A contain zero then \left(\operatorname A\right) \left(\operatorname_X S\right) = \operatorname_X (A S). If S \subseteq X \text S + S \subseteq 2 \operatorname_X S then \operatorname_X S is convex. If R, S \subseteq X then \operatorname_X (R) + \operatorname_X (S) ~\subseteq~ \operatorname_X (R + S)~ \text ~\operatorname_X \left[ \operatorname_X (R) + \operatorname_X (S) \right] ~=~ \operatorname_X (R + S) and so consequently, if R + S is closed then so is \operatorname_X (R) + \operatorname_X (S). If X is a real TVS and S \subseteq X, then \bigcap_ r S \subseteq \operatorname_X S where the left hand side is independent of the topology on X; moreover, if S is a convex neighborhood of the origin then equality holds. For any subset S \subseteq X, \operatorname_X S ~=~ \bigcap_ (S + N) where \mathcal is any neighborhood basis at the origin for X. However, \operatorname_X U ~\supseteq~ \bigcap \ and it is possible for this containment to be proper (for example, if X = \R and S is the rational numbers). It follows that \operatorname_X U \subseteq U + U for every neighborhood U of the origin in X. Closed hulls In a locally convex space, convex hulls of bounded sets are bounded. This is not true for TVSs in general. * The closed convex hull of a set is equal to the closure of the convex hull of that set; that is, equal to \operatorname_X (\operatorname S). * The closed balanced hull of a set is equal to the closure of the balanced hull of that set; that is, equal to \operatorname_X (\operatorname S). * The closed Absolutely convex set, disked hull of a set is equal to the closure of the disked hull of that set; that is, equal to \operatorname_X (\operatorname S). If R, S \subseteq X and the closed convex hull of one of the sets S or R is compact then \operatorname_X (\operatorname (R + S)) ~=~ \operatorname_X (\operatorname R) + \operatorname_X (\operatorname S). If R, S \subseteq X each have a closed convex hull that is compact (that is, \operatorname_X (\operatorname R) and \operatorname_X (\operatorname S) are compact) then \operatorname_X (\operatorname (R \cup S)) ~=~ \operatorname \left[ \operatorname_X (\operatorname R) \cup \operatorname_X (\operatorname S) \right]. Hulls and compactness In a general TVS, the closed convex hull of a compact set may to be compact. The balanced hull of a compact (resp.
totally boundedIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structu ...
) set has that same property. The convex hull of a finite union of compact sets is again compact and convex.


Other properties

Meager, nowhere dense, and Baire A
disk Disk or disc may refer to: * Disk (mathematics) * Disk storage Music * Disc (band), an American experimental music band * Disk (album), ''Disk'' (album), a 1995 EP by Moby Other uses * Disc (galaxy), a disc-shaped group of stars * Disc (magazin ...
in a TVS is not nowhere dense if and only if its closure is a neighborhood of the origin. A vector subspace of a TVS that is closed but not open is nowhere dense. Suppose X is a TVS that does not carry the indiscrete topology. Then X is a Baire space if and only if X has no balanced absorbing nowhere dense subset. A TVS X is a Baire space if and only if X is nonmeager, which happens if and only if there does not exist a nowhere dense set D such that X = \bigcup_ n D. Every nonmeager locally convex TVS is a barrelled space. Important algebraic facts and common misconceptions If S \subseteq X then 2 S \subseteq S + S; if S is convex then equality holds. For an example where equality does hold, let x be non-zero and set S = \; S = \ also works. A subset C is convex if and only if (s + t) C = s C + t C for all positive real s \text t. The disked hull of a set S \subseteq X is equal to the convex hull of the balanced hull of S; that is, equal to \operatorname (\operatorname S). However, in general \operatorname (\operatorname S) ~\neq~ \operatorname (\operatorname S). If R, S \subseteq X and a is a scalar then a(R + S) = aR + a S,~ \text ~\operatorname (R + S) = \operatorname R + \operatorname S,~ \text ~\operatorname (a S) = a \operatorname S. If R, S \subseteq X are convex non-empty disjoint sets and x \not\in R \cup S, then S \cap \operatorname (R \cup \) = \varnothing~ \text ~R \cap \operatorname (S \cup \) = \varnothing. In any non-trivial vector space X, there exist two disjoint non-empty convex subsets whose union is X. Other properties Every TVS topology can be generated by a of ''F''-seminorms.


Properties preserved by set operators

* The balanced hull of a compact (resp.
totally boundedIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structu ...
, open) set has that same property. * The Minkowski sum, (Minkowski) sum of two compact (resp. bounded, balanced, convex) sets has that same property. But the sum of two closed sets need be closed. * The convex hull of a balanced (resp. open) set is balanced (resp. open). However, the convex hull of a closed set need be closed. And the convex hull of a bounded set need be bounded. The following table, the color of each cell indicates whether or not a given property of subsets of X (indicated by the column name, "convex" for instance) is preserved under the set operator (indicated by the row's name, "closure" for instance). If in every TVS, a property is preserved under the indicated set operator then that cell will be colored green; otherwise, it will be colored red. So for instance, since the union of two absorbing sets is again absorbing, the cell in row "R \cup S" and column "Absorbing" is colored green. But since the arbitrary intersection of absorbing sets need not be absorbing, the cell in row "Arbitrary intersections (of at least 1 set)" and column "Absorbing" is colored red. If a cell is not colored then that information has yet to be filled in.


See also

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Notes


Proofs


Citations


Bibliography

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Further reading

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External links

* {{Authority control Topological vector spaces, Topology of function spaces