Topological vector space

In mathematics, 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. 
A topological vector space is a vector space (an algebraic structure) which is also a topological space, this implies that vector space operations be continuous functions. 
More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.
The elements of topological vector spaces are typically functions or linear operators acting on topological vector spaces, and the topology is often defined so as to capture a particular notion of convergence of sequences of functions.
Banach spaces, Hilbert spaces and Sobolev spaces are well-known examples.
Unless stated otherwise, the underlying field of a topological vector space is assumed to be either the complex numbers formula_1 or the real numbers formula_2.
Motivation.
Every normed vector space has a natural topological structure: the norm induces a metric and the metric induces a topology. 
This is a topological vector space because:
Thus all Banach spaces and Hilbert spaces are examples of topological vector 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 functions on an open domain, spaces of infinitely differentiable functions, the Schwartz spaces, and spaces of test functions and the spaces of distributions on them. 
These are all examples of Montel spaces. 
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 criterion.
A topological field is a topological vector space over each of its subfields.
Definition.
A topological vector space (TVS) formula_6 is a vector space over a topological field formula_5 (most often the real or complex numbers with their standard topologies) that is endowed with a topology such that vector addition formula_3 and scalar multiplication formula_10 are continuous functions (where the domains of these functions are endowed with product topologies). Such a topology is called a vector topology or a TVS topology on formula_6.
Every topological vector space is also a commutative topological group under addition.
Some authors (e.g., Walter Rudin) require the topology on formula_6 to be T1; it then follows that the space is Hausdorff, and even Tychonoff. 
A topological vector space is said to be separated 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.
The category of topological vector spaces over a given topological field formula_5 is commonly denoted TVSformula_5 or TVectformula_5. 
The objects are the topological vector spaces over formula_5 and the morphisms are the continuous formula_5-linear maps from one object to another.
A TVS homomorphism or topological homomorphism is a continuous linear map formula_18 between topological vector spaces (TVSs) such that the induced map formula_19 is an open mapping when formula_20, which is the range or image of formula_21, is given the subspace topology induced by "Y".
A TVS embedding or a topological monomorphism is an injective topological homomorphism. Equivalently, a TVS-embedding is a linear map that is also a topological embedding.
A TVS isomorphism or an isomorphism in the category of TVSs is a bijective linear homeomorphism. Equivalently, it is a surjective TVS embedding
Many properties of TVSs that are studied, such as local convexity, metrizability, completeness, and normability, are invariant under TVS isomorphisms.
A collection formula_22 of subsets of a vector space is called additive if for every formula_23, there exists some formula_24 such that formula_25.
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 (i.e. for all formula_26, the map formula_27 defined by formula_28 is a homeomorphism), to define a vector topology it suffices to define a neighborhood basis (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 formula_6 be a vector space and let formula_30 be a sequence of subsets of formula_6. Each set in the sequence formula_32 is called a knot of formula_32 and for every index formula_34, formula_35 is called the formula_34th knot of formula_37 The set formula_38 is called the beginning of formula_37 
The sequence formula_32 is/is a:
If formula_48is an absorbing disk in a vector space formula_6 then the sequence defined by formula_50 forms a string beginning with formula_51. This is called the natural string of formula_48 
Moreover, if a vector space formula_6 has countable dimension then every string contains an absolutely convex string.
Summative sequences of sets have the particularly nice property that they define non-negative continuous real-valued subadditive 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 formula_54 and formula_55 are two collections of subsets of a vector space formula_6 and if formula_57 is a scalar, then by definition:
If formula_68 is a collection sequences of subsets of formula_69 then formula_68 is said to be directed (downwards) under inclusion or simply directed if formula_68 is not empty and for all formula_72 there exists some formula_73 such that formula_74 and formula_75 (said differently, if and only if formula_68 is a prefilter with respect to the containment formula_77 defined above).
Notation: Let formula_78 be the set of all knots of all strings in formula_79
Defining vector topologies using collections of strings is particularly useful for defining classes of TVSs that are not necessarily locally convex.
If formula_68 is the set of all topological strings in a TVS formula_81 then formula_82 
A Hausdorff TVS is metrizable if and only if its topology can be induced by a single topological string.
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. 
Every TVS is completely regular but a TVS need not be normal.
Let formula_6 be a topological vector space. Given a subspace formula_84, the quotient space formula_85 with the usual quotient topology is a Hausdorff topological vector space if and only if formula_86 is closed. 
This permits the following construction: given a topological vector space formula_6 (that is probably not Hausdorff), form the quotient space formula_85 where formula_86 is the closure of formula_90. 
formula_85 is then a Hausdorff topological vector space that can be studied instead of formula_6.
Invariance of vector topologies.
One of the most used properties of vector topologies is that every vector topology is translation invariant:
Scalar multiplication by a non-zero scalar is a TVS-isomorphism. This means that if formula_97 then the linear map formula_27 defined by formula_99 is a homeomorphism. 
Using formula_100 produces the negation map formula_27 defined by formula_102, which is consequently a linear homeomorphism and thus a TVS-isomorphism.
If formula_103 and any subset formula_104, then formula_105 and moreover, if formula_106 then formula_107 is a neighborhood (resp. open neighborhood, closed neighborhood) of formula_108 in formula_6 if and only if the same is true of formula_110 at the origin.
Local notions.
A subset formula_111 of a vector space formula_6 is said to be
Every neighborhood of 0 is an absorbing set and contains an open balanced neighborhood of formula_126 so every topological vector space has a local base of absorbing and balanced sets. 
The origin even has a neighborhood basis consisting of closed balanced neighborhoods of 0; if the space is locally convex then it also has a neighborhood basis consisting of closed convex balanced neighborhoods of 0.
A subset formula_111 of a topological vector space formula_6 is bounded if for every neighborhood formula_129 of the origin, then formula_130 when formula_131 is sufficiently large.
The definition of boundedness can be weakened a bit; formula_111 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, formula_111 is bounded if and only if for every balanced neighborhood formula_129 of 0, there exists formula_131 such that formula_130. 
Moreover, when formula_6 is locally convex, the boundedness can be characterized by seminorms: the subset formula_111 is bounded if and only if every continuous seminorm formula_139 is bounded on formula_111.
Every totally bounded set is bounded. 
If formula_86 is a vector subspace of a TVS formula_6, then a subset of formula_86 is bounded in formula_86 if and only if it is bounded in formula_6.
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 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 formula_126.
Let formula_5 be a non-discrete locally compact topological field, for example the real or complex numbers. 
A Hausdorff topological vector space over formula_5 is locally compact if and only if it is finite-dimensional, that is, isomorphic to formula_149 for some natural number formula_150.
Completeness and uniform structure.
The canonical uniformity on a TVS formula_81 is the unique translation-invariant uniformity that induces the topology formula_152 on formula_6.
Every TVS is assumed to be endowed with this canonical uniformity, which makes all TVSs into uniform spaces. 
This allows one to about related notions such as completeness, uniform convergence, Cauchy nets, and uniform continuity. etc., which are always assumed to be with respect to this uniformity (unless indicated other). 
This implies that every Hausdorff topological vector space is Tychonoff.
A subset of a TVS is compact if and only if it is complete and totally bounded (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).
With respect to this uniformity, a net (or sequence) formula_154 is Cauchy if and only if for every neighborhood formula_129 of formula_126, there exists some index formula_34 such that formula_158 whenever formula_159 and formula_160.
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. 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 linear subspace of a complete topological vector space.
Examples.
Finest and coarsest vector topology.
Let formula_6 be a real or complex vector space.
The trivial topology or indiscrete topology formula_173 is always a TVS topology on any vector space formula_6 and it is the coarsest TVS topology possible. An important consequence of this is that the intersection of any collection of TVS topologies on formula_6 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) complete pseudometrizable seminormable locally convex topological vector space. It is Hausdorff if and only if formula_176.
There exists a TVS topology formula_177 on formula_6 that is finer than every other TVS-topology on formula_6 (that is, any TVS-topology on formula_6 is necessarily a subset of formula_177). Every linear map from formula_182} into another TVS is necessarily continuous. If formula_6 has an uncountable Hamel basis then formula_177 is locally convex and metrizable.
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 formula_6 of all functions formula_186 where formula_2 carries its usual Euclidean topology. This set formula_6 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 formula_189, which carries the natural product topology. With this product topology, formula_190 becomes a topological vector space whose topology is called . The reason for this name is the following: if formula_191 is a sequence (or more generally, a net) of elements in formula_6 and if formula_193 then formula_194 converges to formula_195 in formula_6 if and only if for every real number formula_108, formula_198 converges to formula_199 in formula_2. This TVS is complete, Hausdorff, and locally convex but not metrizable and consequently not normable; indeed, every neighborhood of the origin in the product topology contains lines (i.e. 1-dimensional vector subspaces, which are subsets of the form formula_201 with formula_202).
Finite-dimensional spaces.
Let formula_5 denote formula_2 or formula_1 and endow formula_5 with its usual Hausdorff normed Euclidean topology. Let formula_6 be a vector space over formula_5 of finite dimension formula_209 and so that formula_6 is vector space isomorphic to formula_149 (explicitly, this means that there exists a linear isomorphism between the vector spaces formula_6 and formula_149). This finite-dimensional vector space formula_6 always has a unique vector topology, which makes it TVS-isomorphic to formula_149, where formula_149 is endowed with the usual Euclidean topology (which is the same as the product topology). This Hausdorff vector topology is also the (unique) finest vector topology on formula_6. formula_6 has a unique vector topology if and only if formula_176. If formula_220 then although formula_6 does not have a unique vector topology, it does have a unique vector topology.
The proof of this dichotomy is straightforward so only an outline with the important observations is given. As usual, formula_5 is assumed have the (normed) Euclidean topology. Let formula_6 be a 1-dimensional vector space over formula_5. Observe that if formula_232 is a ball centered at 0 and if formula_104 is a subset containing an "unbounded sequence" then formula_234, where an "unbounded sequence" means a sequence of the form formula_235 where formula_236 and formula_237 is unbounded in normed space formula_5. Any vector topology on formula_6 will be translation invariant and invariant under non-zero scalar multiplication, and for every formula_236, the map formula_241 given by formula_242 is a continuous linear bijection. In particular, for any such formula_108, formula_244 so every subset of formula_6 can be written as formula_246 for some unique subset formula_247 And if this vector topology on formula_6 has a neighborhood of 0 that is properly contained in formula_6, then the continuity of scalar multiplication formula_250 at the origin forces the existence of an open neighborhood of the origin in formula_6 that contain any "unbounded sequence". From this, one deduces that if formula_6 doesn't carry the trivial topology and if formula_236, then for any ball formula_232 center at 0 in formula_5, formula_256 contains an open neighborhood of the origin in formula_6 so that formula_258 is thus a linear homeomorphism. ∎
Non-vector topologies.
If formula_6 is a non-trivial vector space (i.e. of non-zero dimension) then the discrete topology on formula_6 (which is always metrizable) is a TVS topology because despite making addition and negation continuous (which makes it into a topological group under addition), it fails to make scalar multiplication continuous. The cofinite topology on formula_6 (where a subset is open if and only if its complement is finite) is also a TVS topology on formula_6.
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 formula_195 is continuous if formula_282 is bounded (as defined below) for some neighborhood formula_6 of the origin.
A hyperplane on a topological vector space formula_6 is either dense or closed. 
A linear functional formula_195 on a topological vector space formula_6 has either dense or closed kernel. 
Moreover, formula_195 is continuous if and only if its kernel is 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, and the fact that the dual space of the space separates points in the space.
Below are some common topological vector spaces, roughly ordered by their .
Dual space.
Every topological vector space has a continuous dual space—the set formula_301 of all continuous linear functionals, that is, continuous linear maps from the space into the base field formula_302 
A topology on the dual can be defined to be the coarsest topology such that the dual pairing each point evaluation formula_303 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 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 formula_6 is a not-normable locally convex space, then the pairing map formula_305 is never continuous, no matter which vector space topology one chooses on formula_306
Properties.
For any formula_104 of a TVS formula_6, the convex (resp. balanced, disked, closed convex, closed balanced, closed disked) hull of formula_110 is the smallest subset of formula_6 that has this property and contains formula_110.
The closure (resp. interior, convex hull, balanced hull, disked hull) of a set formula_110 is sometimes denoted by formula_313 (resp. formula_314, formula_315, formula_316, formula_317).
Properties preserved by set operators.
The following table, the color of each cell indicates whether or not a given property of subsets of formula_6 (indicated by the column name e.g. "convex") is preserved under the set operator (indicated by the row's name e.g. "closure"). 
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 "formula_543" 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.
 Properties preserved by set operators