Topologies on spaces of linear maps

In mathematics, particularly functional analysis, spaces of linear maps between two vector spaces can be endowed with a variety of topologies. Studying space of linear maps and these topologies can give insight into the spaces themselves.

The article operator topologies discusses topologies on spaces of linear maps between normed spaces, whereas this article discusses topologies on such spaces in the more general setting of topological vector spaces (TVSs).

Topologies of uniform convergence on arbitrary spaces of maps

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Throughout, the following is assumed:

  1.   is any non-empty set and   is a non-empty collection of subsets of   directed by subset inclusion (i.e. for any   there exists some   such that  ).
  2.   is a topological vector space (not necessarily Hausdorff or locally convex).
  3.   is a basis of neighborhoods of 0 in  
  4.   is a vector subspace of  [note 1] which denotes the set of all  -valued functions   with domain  

𝒢-topology

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The following sets will constitute the basic open subsets of topologies on spaces of linear maps. For any subsets   and   let  

The family   forms a neighborhood basis[1] at the origin for a unique translation-invariant topology on   where this topology is not necessarily a vector topology (that is, it might not make   into a TVS). This topology does not depend on the neighborhood basis   that was chosen and it is known as the topology of uniform convergence on the sets in   or as the  -topology.[2] However, this name is frequently changed according to the types of sets that make up   (e.g. the "topology of uniform convergence on compact sets" or the "topology of compact convergence", see the footnote for more details[3]).

A subset   of   is said to be fundamental with respect to   if each   is a subset of some element in   In this case, the collection   can be replaced by   without changing the topology on  [2] One may also replace   with the collection of all subsets of all finite unions of elements of   without changing the resulting  -topology on  [4]

Call a subset   of    -bounded if   is a bounded subset of   for every  [5]

Theorem[2][5] — The  -topology on   is compatible with the vector space structure of   if and only if every   is  -bounded; that is, if and only if for every   and every     is bounded in  

Properties

Properties of the basic open sets will now be described, so assume that   and   Then   is an absorbing subset of   if and only if for all     absorbs  .[6] If   is balanced[6] (respectively, convex) then so is  

The equality   always holds. If   is a scalar then   so that in particular,  [6] Moreover,[4]   and similarly[5]  

For any subsets   and any non-empty subsets  [5]   which implies:

  • if   then  [6]
  • if   then  
  • For any   and subsets   of   if   then  

For any family   of subsets of   and any family   of neighborhoods of the origin in  [4]  

Uniform structure

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For any   and   be any entourage of   (where   is endowed with its canonical uniformity), let   Given   the family of all sets   as   ranges over any fundamental system of entourages of   forms a fundamental system of entourages for a uniform structure on   called the uniformity of uniform converges on   or simply the  -convergence uniform structure.[7] The  -convergence uniform structure is the least upper bound of all  -convergence uniform structures as   ranges over  [7]

Nets and uniform convergence

Let   and let   be a net in   Then for any subset   of   say that   converges uniformly to   on   if for every   there exists some   such that for every   satisfying     (or equivalently,   for every  ).[5]

Theorem[5] — If   and if   is a net in   then   in the  -topology on   if and only if for every     converges uniformly to   on  

Inherited properties

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Local convexity

If   is locally convex then so is the  -topology on   and if   is a family of continuous seminorms generating this topology on   then the  -topology is induced by the following family of seminorms:   as   varies over   and   varies over  .[8]

Hausdorffness

If   is Hausdorff and   then the  -topology on   is Hausdorff.[5]

Suppose that   is a topological space. If   is Hausdorff and   is the vector subspace of   consisting of all continuous maps that are bounded on every   and if   is dense in   then the  -topology on   is Hausdorff.

Boundedness

A subset   of   is bounded in the  -topology if and only if for every     is bounded in  [8]

Examples of 𝒢-topologies

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Pointwise convergence

If we let   be the set of all finite subsets of   then the  -topology on   is called the topology of pointwise convergence. The topology of pointwise convergence on   is identical to the subspace topology that   inherits from   when   is endowed with the usual product topology.

If   is a non-trivial completely regular Hausdorff topological space and   is the space of all real (or complex) valued continuous functions on   the topology of pointwise convergence on   is metrizable if and only if   is countable.[5]

𝒢-topologies on spaces of continuous linear maps

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Throughout this section we will assume that   and   are topological vector spaces.   will be a non-empty collection of subsets of   directed by inclusion.   will denote the vector space of all continuous linear maps from   into   If   is given the  -topology inherited from   then this space with this topology is denoted by  . The continuous dual space of a topological vector space   over the field   (which we will assume to be real or complex numbers) is the vector space   and is denoted by  .

The  -topology on   is compatible with the vector space structure of   if and only if for all   and all   the set   is bounded in   which we will assume to be the case for the rest of the article. Note in particular that this is the case if   consists of (von-Neumann) bounded subsets of  

Assumptions on 𝒢

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Assumptions that guarantee a vector topology

  • (  is directed):   will be a non-empty collection of subsets of   directed by (subset) inclusion. That is, for any   there exists   such that  .

The above assumption guarantees that the collection of sets   forms a filter base. The next assumption will guarantee that the sets   are balanced. Every TVS has a neighborhood basis at 0 consisting of balanced sets so this assumption isn't burdensome.

  • (  are balanced):   is a neighborhoods basis of the origin in   that consists entirely of balanced sets.

The following assumption is very commonly made because it will guarantee that each set   is absorbing in  

  • (  are bounded):   is assumed to consist entirely of bounded subsets of  

The next theorem gives ways in which   can be modified without changing the resulting  -topology on  

Theorem[6] — Let   be a non-empty collection of bounded subsets of   Then the  -topology on   is not altered if   is replaced by any of the following collections of (also bounded) subsets of  :

  1. all subsets of all finite unions of sets in  ;
  2. all scalar multiples of all sets in  ;
  3. all finite Minkowski sums of sets in  ;
  4. the balanced hull of every set in  ;
  5. the closure of every set in  ;

and if   and   are locally convex, then we may add to this list:

  1. the closed convex balanced hull of every set in  

Common assumptions

Some authors (e.g. Narici) require that   satisfy the following condition, which implies, in particular, that   is directed by subset inclusion:

  is assumed to be closed with respect to the formation of subsets of finite unions of sets in   (i.e. every subset of every finite union of sets in   belongs to  ).

Some authors (e.g. Trèves [9]) require that   be directed under subset inclusion and that it satisfy the following condition:

If   and   is a scalar then there exists a   such that  

If   is a bornology on   which is often the case, then these axioms are satisfied. If   is a saturated family of bounded subsets of   then these axioms are also satisfied.

Properties

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Hausdorffness

A subset of a TVS   whose linear span is a dense subset of   is said to be a total subset of   If   is a family of subsets of a TVS   then   is said to be total in   if the linear span of   is dense in  [10]

If   is the vector subspace of   consisting of all continuous linear maps that are bounded on every   then the  -topology on   is Hausdorff if   is Hausdorff and   is total in  [6]

Completeness

For the following theorems, suppose that   is a topological vector space and   is a locally convex Hausdorff spaces and   is a collection of bounded subsets of   that covers   is directed by subset inclusion, and satisfies the following condition: if   and   is a scalar then there exists a   such that  

  •   is complete if
    1.   is locally convex and Hausdorff,
    2.   is complete, and
    3. whenever   is a linear map then   restricted to every set   is continuous implies that   is continuous,
  • If   is a Mackey space then   is complete if and only if both   and   are complete.
  • If   is barrelled then   is Hausdorff and quasi-complete.
  • Let   and   be TVSs with   quasi-complete and assume that (1)   is barreled, or else (2)   is a Baire space and   and   are locally convex. If   covers   then every closed equicontinuous subset of   is complete in   and   is quasi-complete.[11]
  • Let   be a bornological space,   a locally convex space, and   a family of bounded subsets of   such that the range of every null sequence in   is contained in some   If   is quasi-complete (respectively, complete) then so is  .[12]

Boundedness

Let   and   be topological vector spaces and   be a subset of   Then the following are equivalent:[8]

  1.   is bounded in  ;
  2. For every     is bounded in  ;[8]
  3. For every neighborhood   of the origin in   the set   absorbs every  

If   is a collection of bounded subsets of   whose union is total in   then every equicontinuous subset of   is bounded in the  -topology.[11] Furthermore, if   and   are locally convex Hausdorff spaces then

  • if   is bounded in   (that is, pointwise bounded or simply bounded) then it is bounded in the topology of uniform convergence on the convex, balanced, bounded, complete subsets of  [13]
  • if   is quasi-complete (meaning that closed and bounded subsets are complete), then the bounded subsets of   are identical for all  -topologies where   is any family of bounded subsets of   covering  [13]

Examples

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  ("topology of uniform convergence on ...") Notation Name ("topology of...") Alternative name
finite subsets of     pointwise/simple convergence topology of simple convergence
precompact subsets of   precompact convergence
compact convex subsets of     compact convex convergence
compact subsets of     compact convergence
bounded subsets of     bounded convergence strong topology

The topology of pointwise convergence

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By letting   be the set of all finite subsets of     will have the weak topology on   or the topology of pointwise convergence or the topology of simple convergence and   with this topology is denoted by  . Unfortunately, this topology is also sometimes called the strong operator topology, which may lead to ambiguity;[6] for this reason, this article will avoid referring to this topology by this name.

A subset of   is called simply bounded or weakly bounded if it is bounded in  .

The weak-topology on   has the following properties:

  • If   is separable (that is, it has a countable dense subset) and if   is a metrizable topological vector space then every equicontinuous subset   of   is metrizable; if in addition   is separable then so is  [14]
    • So in particular, on every equicontinuous subset of   the topology of pointwise convergence is metrizable.
  • Let   denote the space of all functions from   into   If   is given the topology of pointwise convergence then space of all linear maps (continuous or not)   into   is closed in  .
    • In addition,   is dense in the space of all linear maps (continuous or not)   into  
  • Suppose   and   are locally convex. Any simply bounded subset of   is bounded when   has the topology of uniform convergence on convex, balanced, bounded, complete subsets of   If in addition   is quasi-complete then the families of bounded subsets of   are identical for all  -topologies on   such that   is a family of bounded sets covering  [13]

Equicontinuous subsets

  • The weak-closure of an equicontinuous subset of   is equicontinuous.
  • If   is locally convex, then the convex balanced hull of an equicontinuous subset of   is equicontinuous.
  • Let   and   be TVSs and assume that (1)   is barreled, or else (2)   is a Baire space and   and   are locally convex. Then every simply bounded subset of   is equicontinuous.[11]
  • On an equicontinuous subset   of   the following topologies are identical: (1) topology of pointwise convergence on a total subset of  ; (2) the topology of pointwise convergence; (3) the topology of precompact convergence.[11]

Compact convergence

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By letting   be the set of all compact subsets of     will have the topology of compact convergence or the topology of uniform convergence on compact sets and   with this topology is denoted by  .

The topology of compact convergence on   has the following properties:

  • If   is a Fréchet space or a LF-space and if   is a complete locally convex Hausdorff space then   is complete.
  • On equicontinuous subsets of   the following topologies coincide:
    • The topology of pointwise convergence on a dense subset of  
    • The topology of pointwise convergence on  
    • The topology of compact convergence.
    • The topology of precompact convergence.
  • If   is a Montel space and   is a topological vector space, then   and   have identical topologies.

Topology of bounded convergence

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By letting   be the set of all bounded subsets of     will have the topology of bounded convergence on   or the topology of uniform convergence on bounded sets and   with this topology is denoted by  .[6]

The topology of bounded convergence on   has the following properties:

  • If   is a bornological space and if   is a complete locally convex Hausdorff space then   is complete.
  • If   and   are both normed spaces then the topology on   induced by the usual operator norm is identical to the topology on  .[6]
    • In particular, if   is a normed space then the usual norm topology on the continuous dual space   is identical to the topology of bounded convergence on  .
  • Every equicontinuous subset of   is bounded in  .

Polar topologies

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Throughout, we assume that   is a TVS.

𝒢-topologies versus polar topologies

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If   is a TVS whose bounded subsets are exactly the same as its weakly bounded subsets (e.g. if   is a Hausdorff locally convex space), then a  -topology on   (as defined in this article) is a polar topology and conversely, every polar topology if a  -topology. Consequently, in this case the results mentioned in this article can be applied to polar topologies.

However, if   is a TVS whose bounded subsets are not exactly the same as its weakly bounded subsets, then the notion of "bounded in  " is stronger than the notion of " -bounded in  " (i.e. bounded in   implies  -bounded in  ) so that a  -topology on   (as defined in this article) is not necessarily a polar topology. One important difference is that polar topologies are always locally convex while  -topologies need not be.

Polar topologies have stronger results than the more general topologies of uniform convergence described in this article and we refer the read to the main article: polar topology. We list here some of the most common polar topologies.

List of polar topologies

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Suppose that   is a TVS whose bounded subsets are the same as its weakly bounded subsets.

Notation: If   denotes a polar topology on   then   endowed with this topology will be denoted by   or simply   (e.g. for   we would have   so that   and   all denote   with endowed with  ).

> 
("topology of uniform convergence on ...")
Notation Name ("topology of...") Alternative name
finite subsets of    
 
pointwise/simple convergence weak/weak* topology
 -compact disks   Mackey topology
 -compact convex subsets   compact convex convergence
 -compact subsets
(or balanced  -compact subsets)
  compact convergence
 -bounded subsets  
 
bounded convergence strong topology

𝒢-ℋ topologies on spaces of bilinear maps

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We will let   denote the space of separately continuous bilinear maps and   denote the space of continuous bilinear maps, where   and   are topological vector space over the same field (either the real or complex numbers). In an analogous way to how we placed a topology on   we can place a topology on   and  .

Let   (respectively,  ) be a family of subsets of   (respectively,  ) containing at least one non-empty set. Let   denote the collection of all sets   where     We can place on   the  -topology, and consequently on any of its subsets, in particular on   and on  . This topology is known as the  -topology or as the topology of uniform convergence on the products   of  .

However, as before, this topology is not necessarily compatible with the vector space structure of   or of   without the additional requirement that for all bilinear maps,   in this space (that is, in   or in  ) and for all   and   the set   is bounded in   If both   and   consist of bounded sets then this requirement is automatically satisfied if we are topologizing   but this may not be the case if we are trying to topologize  . The  -topology on   will be compatible with the vector space structure of   if both   and   consists of bounded sets and any of the following conditions hold:

  •   and   are barrelled spaces and   is locally convex.
  •   is a F-space,   is metrizable, and   is Hausdorff, in which case  
  •   and   are the strong duals of reflexive Fréchet spaces.
  •   is normed and   and   the strong duals of reflexive Fréchet spaces.

The ε-topology

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Suppose that   and   are locally convex spaces and let   and   be the collections of equicontinuous subsets of   and  , respectively. Then the  -topology on   will be a topological vector space topology. This topology is called the ε-topology and   with this topology it is denoted by   or simply by  

Part of the importance of this vector space and this topology is that it contains many subspace, such as   which we denote by   When this subspace is given the subspace topology of   it is denoted by  

In the instance where   is the field of these vector spaces,   is a tensor product of   and   In fact, if   and   are locally convex Hausdorff spaces then   is vector space-isomorphic to   which is in turn is equal to  

These spaces have the following properties:

  • If   and   are locally convex Hausdorff spaces then   is complete if and only if both   and   are complete.
  • If   and   are both normed (respectively, both Banach) then so is  

See also

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References

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  1. ^ Because   is just a set that is not yet assumed to be endowed with any vector space structure,   should not yet be assumed to consist of linear maps, which is a notation that currently can not be defined.
  1. ^ Note that each set   is a neighborhood of the origin for this topology, but it is not necessarily an open neighborhood of the origin.
  2. ^ a b c Schaefer & Wolff 1999, pp. 79–88.
  3. ^ In practice,   usually consists of a collection of sets with certain properties and this name is changed appropriately to reflect this set so that if, for instance,   is the collection of compact subsets of   (and   is a topological space), then this topology is called the topology of uniform convergence on the compact subsets of  
  4. ^ a b c Narici & Beckenstein 2011, pp. 19–45.
  5. ^ a b c d e f g h Jarchow 1981, pp. 43–55.
  6. ^ a b c d e f g h i Narici & Beckenstein 2011, pp. 371–423.
  7. ^ a b Grothendieck 1973, pp. 1–13.
  8. ^ a b c d Schaefer & Wolff 1999, p. 81.
  9. ^ Trèves 2006, Chapter 32.
  10. ^ Schaefer & Wolff 1999, p. 80.
  11. ^ a b c d Schaefer & Wolff 1999, p. 83.
  12. ^ Schaefer & Wolff 1999, p. 117.
  13. ^ a b c Schaefer & Wolff 1999, p. 82.
  14. ^ Schaefer & Wolff 1999, p. 87.

Bibliography

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  • Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.
  • Hogbe-Nlend, Henri (1977). Bornologies and Functional Analysis: Introductory Course on the Theory of Duality Topology-Bornology and its use in Functional Analysis. North-Holland Mathematics Studies. Vol. 26. Amsterdam New York New York: North Holland. ISBN 978-0-08-087137-0. MR 0500064. OCLC 316549583.
  • Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
  • Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
  • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
  • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.