Topological vector space
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space which is also a topological space, the latter thereby admitting a notion of continuity.
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.
Hilbert spaces and Banach spaces are well-known examples.
Unless stated otherwise, the underlying field of a topological vector space is assumed to be either the complex numbers ℂ or the real numbers ℝ.
Definition
A topological vector space is a vector space over a topological field ? that is endowed with a topology such that vector addition and scalar multiplication are continuous functions.Some authors require the topology on to be T1; it then follows that the space is Hausdorff, and even Tychonoff.
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 is commonly denoted TVS? or TVect?.
The objects are the topological vector spaces over and the morphisms are the continuous ?-linear maps from one object to another.
Ways of defining a vector topology
If is a vector space and is a topology on, then the addition map is continuous at the origin if and only if the set of neighborhoods of 0 in is additive.This is consequently a necessary condition for a topology to form a vector topology.
Since every vector topology is translation invariant, to define a vector topology it suffices to define a neighborhood basis for it at the origin.
Examples
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:
- The vector addition is jointly continuous with respect to this topology. This follows directly from the triangle inequality obeyed by the norm.
- The scalar multiplication, where is the underlying scalar field of, is jointly continuous. This follows from the triangle inequality and homogeneity of the norm.
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.
If is a vector space then the trivial topology is a TVS topology on.
If is a non-trivial vector space then the discrete topology on is not a TVS topology; furthermore, the cofinite topology on is also not a TVS topology on.
Finest vector topology
Let be a real or complex vector space.There exists a TVS topololgy on that is finer than every other TVS-topology on .
Every linear map from into another TVS is necessarily continuous.
If has an uncountable Hamel basis then is not locally convex and not 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.For instance, the set of all functions : this set can be identified with the product space and carries a natural product topology.
With this topology, becomes a topological vector space, endowed with a topology called the topology of pointwise convergence.
The reason for this name is the following: if is a sequence of elements in, then has limit if and only if has limit for every real number x.
This space is complete, but not normable: indeed, every neighborhood of 0 in the product topology contains lines, i.e., sets for.
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.Hence, every topological vector space is an abelian topological group.
Let be a topological vector space. Given a subspace, the quotient space with the usual quotient topology is a Hausdorff topological vector space if and only if M is closed.
This permits the following construction: given a topological vector space , form the quotient space where M is the closure of.
is then a Hausdorff topological vector space that can be studied instead of.
In particular, topological vector spaces are uniform spaces and one can thus talk about completeness, uniform convergence and uniform continuity.
The vector space operation of addition is uniformly continuous and the scalar multiplication is Cauchy continuous.
Because of this, every topological vector space can be completed and is thus a dense linear subspace of a complete topological vector space.
The Birkhoff–Kakutani theorem gives that the following three conditions on a topological vector space are equivalent:
- The origin 0 is closed in, and there is a countable basis of neighborhoods for 0 in.
- is metrizable.
- There is a translation-invariant metric on that induces the given topology on.
By the Birkhoff–Kakutani theorem, it follows that there is an equivalent metric that is translation-invariant.
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 0.
A linear operator between two topological vector spaces which is continuous at one point is continuous on the whole domain.
Moreover, a linear operator is continuous if is bounded for some neighborhood of 0.
A hyperplane on a topological vector space is either dense or closed. A linear functional on a topological vector space has either dense or closed kernel.
Moreover, is continuous if and only if its kernel is closed.
Let be a non-discrete locally compact topological field, for example the real or complex numbers.
A topological vector space over is locally compact if and only if it is finite-dimensional, that is, isomorphic to for some natural number.
Local notions
A subset of a topological vector space is said to be- symmetric if, or equivalently, if ;
- balanced if for every scalar
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 niceness.- F-spaces are complete topological vector spaces with a translation-invariant metric. These include spaces for all.
- Locally convex topological vector spaces: here each point has a local base consisting of convex sets. By a technique known as Minkowski functionals it can be shown that a space is locally convex if and only if its topology can be defined by a family of semi-norms. Local convexity is the minimum requirement for "geometrical" arguments like the Hahn–Banach theorem. The spaces are locally convex for all, but not for.
- 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 reflexivity condition, where the dual space is endowed with the topology of uniform convergence on totally bounded sets.
- Montel space: a barrelled space where every closed and bounded set is 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 semi-norms. Many interesting spaces of functions fall into this class. A locally convex F-space is a Fréchet space.
- LF-spaces are 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 semi-normed spaces: locally convex spaces where the topology can be described by a single norm or semi-norm. In normed spaces a linear operator is continuous if and only if it is bounded.
- Banach spaces: Complete normed vector spaces. Most of functional analysis is formulated for Banach spaces.
- Reflexive Banach spaces: Banach spaces naturally isomorphic to their double dual, which ensures that some geometrical arguments can be carried out. An important example which is not reflexive is Lp space|, whose dual is but is strictly contained in the dual of.
- Hilbert spaces: these have an inner product; even though these spaces may be infinite-dimensional, most geometrical reasoning familiar from finite dimensions can be carried out in them. These include spaces.
- Euclidean spaces: or with the topology induced by the standard inner product. As pointed out in the preceding section, for a given finite, 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.
Dual space
A topology on the dual can be defined to be the coarsest topology such that the dual pairing each point evaluation 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.
Caution: Whenever is a not-normable locally convex space, then the pairing map is never continuous, no matter which vector space topology one chooses on V*.Properties
Let be a TVS.
Every TVS is a commutative topological group.
We denote the closure of a set by .
;Translation invariant topology
One of the most used properties of vector topologies is that every vector topology is translation invariant:
Thus for any and any subset, is a Neighborhood neighborhood of if and only if the same is true of at the origin.
The map defined by is also a homeomorphism.
;Properties of neighborhoods and open sets- The open convex subsets of a TVS are exactly those that are of the form for some and some positive continuous sublinear functional on.
- If and is an open subset of then is an open set in.
- Every neighborhood of 0 is an absorbing set and contains an open balanced neighborhood of 0.
- The origin 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.
- If has non-empty interior then is a neighborhood of 0.
- If is an absorbing disk in a TVS and if is the Minkowski functional of then
- Note that we did not assume that had any topological properties nor that was continuous.
- A disk in a TVS is not nowhere dense if and only if its closure is a neighborhood of the origin.
- Any connected open subset of a TVS is arcwise connected.
- Let and be two vector topologies on. Then if and only if whenever a net in converges 0 in then in.
- Let be a neighborhood basis of the origin in, let, and let. Then if and only if there exists a net in such that in.
- 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.
- If has non-empty interior then and.
- If and has non-empty interior then.
- If is a disk in that has non-empty interior then 0 belongs to the interior of.
- However, a closed balanced subset of with non-empty interior may fail to contain 0 in its interior.
- If the interior of a balanced set is non-empty but does not contain the origin then it this interior can not be a balanced set.
- Every TVS has a completion and every Hausdorff TVS has a Hausdorff completion.
- A compact subset of a TVS is complete.
- If a TVS has a complete neighborhood of the origin then it is complete.
- A complete subset of a Hausdorff TVS is closed.
- If is a complete subset of a TVS then any subset of that is closed in is complete.
- If and is a scalar then ; if is Hausdorff,, or then equality holds:.
- Thus, every non-zero scalar multiple of a closed set is closed. If is Hausdorff then every scalar multiple of a closed set is closed.
- In particular, every neighborhood of the origin contains the closure of.
- It follows that every subset of that contains the origin is compact and thus complete. In particular, if is not Hausdorff then there exist compact complete sets that are not closed.
- Note however, that there are sets in TVSs such that.
- The closed convex hull of a set is equal to the closure of the convex hull of that set.
- The closed balanced hull of a set is equal to the closure of the balanced hull of that set.
- The closed disked hull of a set is equal to the closure of the disked hull of that set.
- If and the closed convex hull of one of the sets or is compact then.
- If each have a closed convex hull that is compact then.
- A subset of a TVS is compact if and only if it is complete and totally bounded.
- Thus, in a complete TVS, a closed and totally bounded subset is compact.
In a general TVS, the closed convex hull of a compact set may fail to be compact.
- The balanced hull of a compact set has that same property.
- In a Fréchet space, the closed convex hull of a compact set is compact.
- The convex hull of a finite union of compact convex sets is again compact and convex.
- In a locally convex space, the convex hull and the disked hull of a totally bounded set is totally bounded.
- In a complete locally convex space, the convex hull and the disked hull of a compact set are both compact.
- More generally, if is a compact subset of a locally convex space, then the convex hull is compact if and only if it is complete.
- Every totally bounded set is bounded.
- A set is bounded if and only if each of its subsequences is a bounded set.
- A vector subspace of a TVS is bounded if and only if it is contained in the closure of.
- In a locally convex space, convex hulls of bounded sets are bounded. This is not true for TVSs in general.
- If is a vector subspace of a TVS, then a subset of is bounded in if and only if it is bounded in.
- A series is said to converge in a TVS if the sequence of partial sums converges.
- If a series converges in a TVS then in.
- If then ; if is convex then equality holds.
- For an example where equality does not hold, let be non-zero and set ; also works.
- However, in general.
- Every TVS is completely regular. However, a TVS need not be normal.
- Every TVS is connected.
- 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.
- Every TVS topology can be generated by a family of F-seminorms.
- Suppose is a TVS that does not carry the indiscrete topology. Then is a Baire space if and only if has no balanced absorbing nowhere dense subset.
- A TVS is a Baire space if and only if is nonmeager, which happens if and only if there does not exist a nowhere dense set such that.
- Note that every nonmeager locally convex TVS is a barrelled space.
Properties preserved by set operators
- The balanced hull of a compact set has that same property.
- The interior of a convex set is convex.
- The closure of a convex set is has that same property.
- The closure of a vector subspace is a vector subspace.
- The sum of two compact sets has that same property.
- The sum of two closed sets need not be closed.
- The convex hull of a closed set need not be closed.
- The convex hull of a bounded set need not be bounded.
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 "" and column "Absorbing" is colored green.
But since the arbitrary intersection of absorbing sets need not be absorbing, the cell in row "Arbitrary intersections " 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