Dual space topology of uniform convergence on some sub-collection of bounded subsets
In functional analysis and related areas of mathematics a polar topology, topology of
-convergence or topology of uniform convergence on the sets of
is a method to define locally convex topologies on the vector spaces of a pairing.
Preliminaries
A pairing is a triple
consisting of two vector spaces over a field
(either the real numbers or complex numbers) and a bilinear map
A dual pair or dual system is a pairing
satisfying the following two separation axioms:
separates/distinguishes points of
: for all non-zero
there exists
such that
and
separates/distinguishes points of
: for all non-zero
there exists
such that ![{\displaystyle b(x,y)\neq 0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e33251dee6a480032a05f177786766a4fecd8b6e)
Polars
The polar or absolute polar of a subset
is the set
![{\displaystyle A^{\circ }:=\left\{y\in Y:\sup _{x\in A}|b(x,y)|\leq 1\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68c63e2282b6d38b1e152a9fea5435999e613680)
Dually, the polar or absolute polar of a subset
is denoted by
and defined by
![{\displaystyle B^{\circ }:=\left\{x\in X:\sup _{y\in B}|b(x,y)|\leq 1\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c49953d293b1c140d273e50ecef34eb24aa8561d)
In this case, the absolute polar of a subset
is also called the prepolar of
and may be denoted by
The polar is a convex balanced set containing the origin.
If
then the bipolar of
denoted by
is defined by
Similarly, if
then the bipolar of
is defined to be
Weak topologies
Suppose that
is a pairing of vector spaces over
- Notation: For all
let
denote the linear functional on
defined by
and let ![{\displaystyle b(X,\bullet )=\left\{b(x,\bullet )~:~x\in X\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa6bf505b68669ab7fe12c266995caed29853fc3)
- Similarly, for all
let
be defined by
and let ![{\displaystyle b(\bullet ,Y)=\left\{b(\bullet ,y)~:~y\in Y\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/947c6268cbfb135ab980a8f851c8b0cc78693441)
The weak topology on
induced by
(and
) is the weakest TVS topology on
denoted by
or simply
making all maps
continuous, as
ranges over
Similarly, there are the dual definition of the weak topology on
induced by
(and
), which is denoted by
or simply
: it is the weakest TVS topology on
making all maps
continuous, as
ranges over
Weak boundedness and absorbing polars
It is because of the following theorem that it is almost always assumed that the family
consists of
-bounded subsets of
Theorem — For any subset
the following are equivalent:
is an absorbing subset of
- If this condition is not satisfied then
can not possibly be a neighborhood of the origin in any TVS topology on
;
is a
-bounded set; said differently,
is a bounded subset of
; - for all
where this supremum may also be denoted by ![{\displaystyle ~\sup \left|b(A,y)\right|.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43ebd08147ae75814e3773e3455a702da9857f08)
The
-bounded subsets of
have an analogous characterization.
Dual definitions and results
Every pairing
can be associated with a corresponding pairing
where by definition
There is a repeating theme in duality theory, which is that any definition for a pairing
has a corresponding dual definition for the pairing
- Convention and Definition: Given any definition for a pairing
one obtains a dual definition by applying it to the pairing
If the definition depends on the order of
and
(e.g. the definition of "the weak topology
defined on
by
") then by switching the order of
and
it is meant that this definition should be applied to
(e.g. this gives us the definition of "the weak topology
defined on
by
").
For instance, after defining "
distinguishes points of
" (resp, "
is a total subset of
") as above, then the dual definition of "
distinguishes points of
" (resp, "
is a total subset of
") is immediately obtained. For instance, once
is defined then it should be automatically assume that
has been defined without mentioning the analogous definition. The same applies to many theorems.
- Convention: Adhering to common practice, unless clarity is needed, whenever a definition (or result) is given for a pairing
then mention the corresponding dual definition (or result) will be omitted but it may nevertheless be used.
In particular, although this article will only define the general notion of polar topologies on
with
being a collection of
-bounded subsets of
this article will nevertheless use the dual definition for polar topologies on
with
being a collection of
-bounded subsets of
- Identification of
with ![{\displaystyle (Y,X)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a5ac08933cce64f0272b438687034d753bfc627)
Although it is technically incorrect and an abuse of notation, the following convention is nearly ubiquitous:
- Convention: This article will use the common practice of treating a pairing
interchangeably with
and also denoting
by ![{\displaystyle (Y,X,b).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55238264f504d5312a6c1d321643093ede71461b)
Polar topologies
Throughout,
is a pairing of vector spaces over the field
and
is a non-empty collection of
-bounded subsets of
For every
and
is convex and balanced and because
is a
-bounded, the set
is absorbing in
The polar topology on
determined (or generated) by
(and
), also called the
-topology on
or the topology of uniform convergence on the sets of
is the unique topological vector space (TVS) topology on
for which
![{\displaystyle \left\{rG^{\circ }~:~G\in {\mathcal {G}},r>0\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c21c1558559fe8341fd85d52de42897127636371)
forms a neighbourhood subbasis at the origin. When
is endowed with this
-topology then it is denoted by
If
is a sequence of positive numbers converging to
then the defining neighborhood subbasis at
may be replaced with
![{\displaystyle \left\{r_{i}G^{\circ }~:~G\in {\mathcal {G}},i=1,2,\ldots \right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cbfdc9a3ac0b18c71180154be64d8caac8fc5b5)
without changing the resulting topology.
When
is a directed set with respect to subset inclusion (i.e. if for all
there exists some
such that
) then the defining neighborhood subbasis at the origin actually forms a neighborhood basis at
- Seminorms defining the polar topology
Every
determines a seminorm
defined by
![{\displaystyle p_{G}(y)=\sup _{g\in G}|b(g,y)|=\sup |b(G,y)|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fe36e683162d8d5b29af7db352c00d4e9b7c49c)
where
and
is in fact the Minkowski functional of
Because of this, the
-topology on
is always a locally convex topology.
- Modifying
![{\displaystyle {\mathcal {G}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a980c59d42c003fd07fdf3646e1fb95ff82f99)
If every positive scalar multiple of a set in
is contained in some set belonging to
then the defining neighborhood subbasis at the origin can be replaced with
![{\displaystyle \left\{G^{\circ }:G\in {\mathcal {G}}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/122cd2ccd0b6128dfd6889f3dae41221893a8c02)
without changing the resulting topology.
The following theorem gives ways in which
can be modified without changing the resulting
-topology on
Theorem — Let
is a pairing of vector spaces over
and let
be a non-empty collection of
-bounded subsets of
The
-topology on
is not altered if
is replaced by any of the following collections of [
-bounded] subsets of
:
- all subsets of all finite unions of sets in
; - all scalar multiples of all sets in
; - the balanced hull of every set in
; - the convex hull of every set in
; - the
-closure of every set in
; - the
-closure of the convex balanced hull of every set in ![{\displaystyle {\mathcal {G}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00c34bd8de01a2dca547f2c8f7dfaa4390e02738)
It is because of this theorem that many authors often require that
also satisfy the following additional conditions:
- The union of any two sets
is contained in some set
; - All scalar multiples of every
belongs to ![{\displaystyle {\mathcal {G}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00c34bd8de01a2dca547f2c8f7dfaa4390e02738)
Some authors[4] further assume that every
belongs to some set
because this assumption suffices to ensure that the
-topology is Hausdorff.
- Convergence of nets and filters
If
is a net in
then
in the
-topology on
if and only if for every
or in words, if and only if for every
the net of linear functionals
on
converges uniformly to
on
; here, for each
the linear functional
is defined by
If
then
in the
-topology on
if and only if for all
A filter
on
converges to an element
in the
-topology on
if
converges uniformly to
on each
Properties
- The results in the article Topologies on spaces of linear maps can be applied to polar topologies.
Throughout,
is a pairing of vector spaces over the field
and
is a non-empty collection of
-bounded subsets of
- Hausdorffness
- We say that
covers
if every point in
belong to some set in ![{\displaystyle {\mathcal {G}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00c34bd8de01a2dca547f2c8f7dfaa4390e02738)
- We say that
is total in
if the linear span of
is dense in ![{\displaystyle X.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03)
Theorem — Let
be a pairing of vector spaces over the field
and
be a non-empty collection of
-bounded subsets of
Then,
- If
covers
then the
-topology on
is Hausdorff. - If
distinguishes points of
and if
is a
-dense subset of
then the
-topology on
is Hausdorff. - If
is a dual system (rather than merely a pairing) then the
-topology on
is Hausdorff if and only if span of
is dense in ![{\displaystyle (X,\sigma (X,Y,b)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7db02f0cca9387bfb1e3ebc1536788f2b334b9d)
Proof |
Proof of (2): If then we're done, so assume otherwise. Since the -topology on is a TVS topology, it suffices to show that the set is closed in Let be non-zero, let be defined by for all and let Since distinguishes points of there exists some (non-zero) such that where (since is surjective) it can be assumed without loss of generality that The set is a -open subset of that is not empty (since it contains ). Since is a -dense subset of there exists some and some such that Since so that where is a subbasic closed neighborhood of the origin in the -topology on ■ |
Examples of polar topologies induced by a pairing
Throughout,
will be a pairing of vector spaces over the field
and
will be a non-empty collection of
-bounded subsets of
The following table will omit mention of
The topologies are listed in an order that roughly corresponds with coarser topologies first and the finer topologies last; note that some of these topologies may be out of order e.g.
and the topology below it (i.e. the topology generated by
-complete and bounded disks) or if
is not Hausdorff. If more than one collection of subsets appears the same row in the left-most column then that means that the same polar topology is generated by these collections.
- Notation: If
denotes a polar topology on
then
endowed with this topology will be denoted by
or simply
For example, if
then
so that
and
all denote
with endowed with ![{\displaystyle \sigma (X,Y,b).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4f57985c6562456109758d315c36c5eb685c44c)
![{\displaystyle {\mathcal {G}}\subseteq \wp (X)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf8bed8958ddbcb45fe655fa5ffc8352943605d6) ("topology of uniform convergence on ...") | Notation | Name ("topology of...") | Alternative name |
finite subsets of ![{\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab) (or -closed disked hulls of finite subsets of ) | ![{\displaystyle \sigma (X,Y,b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ad3b8a1188a203e79c49539eff3cb417c9720a5)
| pointwise/simple convergence | weak/weak* topology |
-compact disks | | | Mackey topology |
-compact convex subsets | | compact convex convergence | |
-compact subsets (or balanced -compact subsets) | | compact convergence | |
-complete and bounded disks | | convex balanced complete bounded convergence | |
-precompact/totally bounded subsets (or balanced -precompact subsets) | | precompact convergence | |
-infracomplete and bounded disks | | convex balanced infracomplete bounded convergence | |
-bounded subsets | ![{\displaystyle b(X,Y,b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/707033176b61470aa238662bb9ea78887db4d18f)
| bounded convergence | strong topology Strongest polar topology |
Weak topology σ(Y, X)
For any
a basic
-neighborhood of
in
is a set of the form:
![{\displaystyle \left\{z\in X:|b(z-x,y_{i})|\leq r{\text{ for all }}i\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/945b4c533f9d6269eacd0dfbf02bbfed21b6b4e8)
for some real
and some finite set of points
in
The continuous dual space of
is
where more precisely, this means that a linear functional
on
belongs to this continuous dual space if and only if there exists some
such that
for all
The weak topology is the coarsest TVS topology on
for which this is true.
In general, the convex balanced hull of a
-compact subset of
need not be
-compact.
If
and
are vector spaces over the complex numbers (which implies that
is complex valued) then let
and
denote these spaces when they are considered as vector spaces over the real numbers
Let
denote the real part of
and observe that
is a pairing. The weak topology
on
is identical to the weak topology
This ultimately stems from the fact that for any complex-valued linear functional
on
with real part
then
for all ![{\displaystyle y\in Y.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b14bc7ecf2f86320b4a930f4961919ae45a34d9c)
Mackey topology τ(Y, X)
The continuous dual space of
is
(in the exact same way as was described for the weak topology). Moreover, the Mackey topology is the finest locally convex topology on
for which this is true, which is what makes this topology important.
Since in general, the convex balanced hull of a
-compact subset of
need not be
-compact, the Mackey topology may be strictly coarser than the topology
Since every
-compact set is
-bounded, the Mackey topology is coarser than the strong topology
Strong topology 𝛽(Y, X)
A neighborhood basis (not just a subbasis) at the origin for the
topology is:
![{\displaystyle \left\{A^{\circ }~:~A\subseteq X{\text{ is a }}\sigma (X,Y,b)-{\text{bounded}}{\text{ subset of }}X\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83cfd605b6d7e3aa2f1c652e78353fcd4e3ef4d5)
The strong topology
is finer than the Mackey topology.
Polar topologies and topological vector spaces
Throughout this section,
will be a topological vector space (TVS) with continuous dual space
and
will be the canonical pairing, where by definition
The vector space
always distinguishes/separates the points of
but
may fail to distinguishes the points of
(this necessarily happens if, for instance,
is not Hausdorff), in which case the pairing
is not a dual pair. By the Hahn–Banach theorem, if
is a Hausdorff locally convex space then
separates points of
and thus
forms a dual pair.
Properties
- If
covers
then the canonical map from
into
is well-defined. That is, for all
the evaluation functional on
meaning the map
is continuous on
- If in addition
separates points on
then the canonical map of
into
is an injection.
- Suppose that
is a continuous linear and that
and
are collections of bounded subsets of
and
respectively, that each satisfy axioms
and
Then the transpose of
is continuous if for every
there is some
such that
- In particular, the transpose of
is continuous if
carries the
(respectively,
) topology and
carry any topology stronger than the
topology (respectively,
).
- If
is a locally convex Hausdorff TVS over the field
and
is a collection of bounded subsets of
that satisfies axioms
and
then the bilinear map
defined by
is continuous if and only if
is normable and the
-topology on
is the strong dual topology ![{\displaystyle b(X',X).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e6075f1adde1625910314224c8eb2a2c196906f)
- Suppose that
is a Fréchet space and
is a collection of bounded subsets of
that satisfies axioms
and
If
contains all compact subsets of
then
is complete.
Polar topologies on the continuous dual space
Throughout,
will be a TVS over the field
with continuous dual space
and
and
will be associated with the canonical pairing. The table below defines many of the most common polar topologies on
- Notation: If
denotes a polar topology then
endowed with this topology will be denoted by
(e.g. if
then
and
so that
denotes
with endowed with
).
If in addition,
then this TVS may be denoted by
(for example,
).
![{\displaystyle {\mathcal {G}}\subseteq \wp (X)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf8bed8958ddbcb45fe655fa5ffc8352943605d6) ("topology of uniform convergence on ...") | Notation | Name ("topology of...") | Alternative name |
finite subsets of ![{\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab) (or -closed disked hulls of finite subsets of ) | ![{\displaystyle \sigma (X',X)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93b21e0d932fc99479e02db1902a890cbb700b55)
| pointwise/simple convergence | weak/weak* topology |
compact convex subsets | | compact convex convergence | |
compact subsets (or balanced compact subsets) | | compact convergence | |
-compact disks | | | Mackey topology |
precompact/totally bounded subsets (or balanced precompact subsets) | | precompact convergence | |
complete and bounded disks | | convex balanced complete bounded convergence | |
infracomplete and bounded disks | | convex balanced infracomplete bounded convergence | |
bounded subsets | ![{\displaystyle b(X',X)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99936b37a01683c8b41ace7c88a88412c6f7daea)
| bounded convergence | strong topology |
-compact disks in | | | Mackey topology |
The reason why some of the above collections (in the same row) induce the same polar topologies is due to some basic results. A closed subset of a complete TVS is complete and that a complete subset of a Hausdorff and complete TVS is closed. Furthermore, in every TVS, compact subsets are complete and the balanced hull of a compact (resp. totally bounded) subset is again compact (resp. totally bounded). Also, a Banach space can be complete without being weakly complete.
If
is bounded then
is absorbing in
(note that being absorbing is a necessary condition for
to be a neighborhood of the origin in any TVS topology on
). If
is a locally convex space and
is absorbing in
then
is bounded in
Moreover, a subset
is weakly bounded if and only if
is absorbing in
For this reason, it is common to restrict attention to families of bounded subsets of
Weak/weak* topology σ(X', X)
The
topology has the following properties:
- Banach–Alaoglu theorem: Every equicontinuous subset of
is relatively compact for
- it follows that the
-closure of the convex balanced hull of an equicontinuous subset of
is equicontinuous and
-compact.
- Theorem (S. Banach): Suppose that
and
are Fréchet spaces or that they are duals of reflexive Fréchet spaces and that
is a continuous linear map. Then
is surjective if and only if the transpose of
is one-to-one and the image of
is weakly closed in ![{\displaystyle X'_{\sigma (X',X)}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89f72d7963db630dc13f8c59b59f5f0013a1c1d4)
- Suppose that
and
are Fréchet spaces,
is a Hausdorff locally convex space and that
is a separately-continuous bilinear map. Then
is continuous. - In particular, any separately continuous bilinear maps from the product of two duals of reflexive Fréchet spaces into a third one is continuous.
is normable if and only if
is finite-dimensional. - When
is infinite-dimensional the
topology on
is strictly coarser than the strong dual topology ![{\displaystyle b(X',X).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e6075f1adde1625910314224c8eb2a2c196906f)
- Suppose that
is a locally convex Hausdorff space and that
is its completion. If
then
is strictly finer than ![{\displaystyle \sigma (X',X).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fb123efd4327ec07d7f217f1192aed04761909d)
- Any equicontinuous subset in the dual of a separable Hausdorff locally convex vector space is metrizable in the
topology. - If
is locally convex then a subset
is
-bounded if and only if there exists a barrel
in
such that ![{\displaystyle H\subseteq B^{\circ }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d45ee3c34d61d3f99e10cdbdd3e6546d116aa40e)
Compact-convex convergence γ(X', X)
If
is a Fréchet space then the topologies
Compact convergence c(X', X)
If
is a Fréchet space or a LF-space then
is complete.
Suppose that
is a metrizable topological vector space and that
If the intersection of
with every equicontinuous subset of
is weakly-open, then
is open in
Precompact convergence
Banach–Alaoglu theorem: An equicontinuous subset
has compact closure in the topology of uniform convergence on precompact sets. Furthermore, this topology on
coincides with the
topology.
Mackey topology τ(X', X)
By letting
be the set of all convex balanced weakly compact subsets of
will have the Mackey topology on
or the topology of uniform convergence on convex balanced weakly compact sets, which is denoted by
and
with this topology is denoted by
Strong dual topology b(X', X)
Due to the importance of this topology, the continuous dual space of
is commonly denoted simply by
Consequently,
The
topology has the following properties:
- If
is locally convex, then this topology is finer than all other
-topologies on
when considering only
's whose sets are subsets of ![{\displaystyle X.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03)
- If
is a bornological space (e.g. metrizable or LF-space) then
is complete. - If
is a normed space then the strong dual topology on
may be defined by the norm
where ![{\displaystyle x'\in X'.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6779232214814af1e66caf46930b7bc0002507a0)
- If
is a LF-space that is the inductive limit of the sequence of space
(for
) then
is a Fréchet space if and only if all
are normable. - If
is a Montel space then
has the Heine–Borel property (i.e. every closed and bounded subset of
is compact in
) - On bounded subsets of
the strong and weak topologies coincide (and hence so do all other topologies finer than
and coarser than
). - Every weakly convergent sequence in
is strongly convergent.
Mackey topology τ(X, X'')
By letting
be the set of all convex balanced weakly compact subsets of
will have the Mackey topology on
induced by
or the topology of uniform convergence on convex balanced weakly compact subsets of
, which is denoted by
and
with this topology is denoted by
- This topology is finer than
and hence finer than ![{\displaystyle \tau (X',X).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c842489f9462f8e7fce7f06ecb85cb098a591c08)
Polar topologies induced by subsets of the continuous dual space
Throughout,
will be a TVS over the field
with continuous dual space
and the canonical pairing will be associated with
and
The table below defines many of the most common polar topologies on
- Notation: If
denotes a polar topology on
then
endowed with this topology will be denoted by
or
(e.g. for
we'd have
so that
and
both denote
with endowed with
).
![{\displaystyle {\mathcal {G}}\subseteq \wp (X)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf8bed8958ddbcb45fe655fa5ffc8352943605d6) ("topology of uniform convergence on ...") | Notation | Name ("topology of...") | Alternative name |
finite subsets of ![{\displaystyle X'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/865f8505e90120a535a4ee68ca253dbd8ce7eb6a) (or -closed disked hulls of finite subsets of ) | ![{\displaystyle \sigma \left(X,X'\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e8700cb58b5c4e1d52a2c440292ad11bb9f9fda) | pointwise/simple convergence | weak topology |
equicontinuous subsets (or equicontinuous disks) (or weak-* compact equicontinuous disks) | | equicontinuous convergence | |
weak-* compact disks | | | Mackey topology |
weak-* compact convex subsets | | compact convex convergence | |
weak-* compact subsets (or balanced weak-* compact subsets) | | compact convergence | |
weak-* bounded subsets | ![{\displaystyle b\left(X,X'\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d46ada9ee9d7b126873f6643973f650ea17118a)
| bounded convergence | strong topology |
The closure of an equicontinuous subset of
is weak-* compact and equicontinuous and furthermore, the convex balanced hull of an equicontinuous subset is equicontinuous.
Weak topology 𝜎(X, X')
Suppose that
and
are Hausdorff locally convex spaces with
metrizable and that
is a linear map. Then
is continuous if and only if
is continuous. That is,
is continuous when
and
carry their given topologies if and only if
is continuous when
and
carry their weak topologies.
Convergence on equicontinuous sets 𝜀(X, X')
If
was the set of all convex balanced weakly compact equicontinuous subsets of
then the same topology would have been induced.
If
is locally convex and Hausdorff then
's given topology (i.e. the topology that
started with) is exactly
That is, for
Hausdorff and locally convex, if
then
is equicontinuous if and only if
is equicontinuous and furthermore, for any
is a neighborhood of the origin if and only if
is equicontinuous.
Importantly, a set of continuous linear functionals
on a TVS
is equicontinuous if and only if it is contained in the polar of some neighborhood
of the origin in
(i.e.
). Since a TVS's topology is completely determined by the open neighborhoods of the origin, this means that via operation of taking the polar of a set, the collection of equicontinuous subsets of
"encode" all information about
's topology (i.e. distinct TVS topologies on
produce distinct collections of equicontinuous subsets, and given any such collection one may recover the TVS original topology by taking the polars of sets in the collection). Thus uniform convergence on the collection of equicontinuous subsets is essentially "convergence on the topology of
".
Mackey topology τ(X, X')
Suppose that
is a locally convex Hausdorff space. If
is metrizable or barrelled then
's original topology is identical to the Mackey topology
Topologies compatible with pairings
Let
be a vector space and let
be a vector subspace of the algebraic dual of
that separates points on
If
is any other locally convex Hausdorff topological vector space topology on
then
is compatible with duality between
and
if when
is equipped with
then it has
as its continuous dual space. If
is given the weak topology
then
is a Hausdorff locally convex topological vector space (TVS) and
is compatible with duality between
and
(i.e.
). The question arises: what are all of the locally convex Hausdorff TVS topologies that can be placed on
that are compatible with duality between
and
? The answer to this question is called the Mackey–Arens theorem.
See also
References
- Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
- Robertson, A.P.; Robertson, W. (1964). Topological vector spaces. Cambridge University Press.
- 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.
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