In mathematics, a linear map is a mapping
V → W between two modules (including vector
spaces) that preserves the operations of addition
and scalar multiplication.
By studying the linear maps between two modules
one can gain insight into their structures.
If the modules have additional structure,
like topologies or bornologies, then one can
study the subspace of linear maps that preserve
this structure.
== Topologies of uniform convergence ==
Suppose that T is any set and that
G
{\displaystyle {\mathcal {G}}}
is a collection of subsets of T directed by
inclusion. Suppose in addition that Y is a
topological vector space (not necessarily
Hausdorff or locally convex) and that
N
{\displaystyle {\mathcal {N}}}
is a basis of neighborhoods of 0 in Y. Then
the set of all functions from T into Y,
Y
T
{\displaystyle Y^{T}}
, can be given a unique translation-invariant
topology by defining a basis of neighborhoods
of 0 in
Y
T
{\displaystyle Y^{T}}
, to be
U
(
G
,
N
)
=
{
f
∈
Y
T
:
f
(
G
)
⊆
N
}
{\displaystyle {\mathcal {U}}(G,N)=\{f\in
Y^{T}:f(G)\subseteq N\}}
as G and N range over all
G
∈
G
{\displaystyle G\in {\mathcal {G}}}
and
N
∈
N
{\displaystyle N\in {\mathcal {N}}}
. This topology does not depend on the basis
N
{\displaystyle {\mathcal {N}}}
that was chosen and it is known as the topology
of uniform convergence on the sets in
G
{\displaystyle {\mathcal {G}}}
or as the
G
{\displaystyle {\mathcal {G}}}
-topology. In practice,
G
{\displaystyle {\mathcal {G}}}
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,
G
{\displaystyle {\mathcal {G}}}
is the collection of compact subsets of T
(and T is a topological space), then this
topology is called the topology of uniform
convergence on the compact subsets of T. A
set
G
1
{\displaystyle {\mathcal {G}}_{1}}
of
G
{\displaystyle {\mathcal {G}}}
is said to be fundamental with respect to
G
{\displaystyle {\mathcal {G}}}
if each
G
∈
G
{\displaystyle G\in {\mathcal {G}}}
is a subset of some element in
G
1
{\displaystyle {\mathcal {G}}_{1}}
. In this case, the collection
G
{\displaystyle {\mathcal {G}}}
can be replaced by
G
1
{\displaystyle {\mathcal {G}}_{1}}
without changing the topology on
Y
T
{\displaystyle Y^{T}}
.However, the
G
{\displaystyle {\mathcal {G}}}
-topology on
Y
T
{\displaystyle Y^{T}}
is not necessarily compatible with the vector
space structure of
Y
T
{\displaystyle Y^{T}}
or of any of its vector subspaces (that is,
it is not necessarily a topological vector
space topology on
Y
T
{\displaystyle Y^{T}}
). Suppose that F is a vector subspace
Y
T
{\displaystyle Y^{T}}
so that it inherits the subspace topology
from
Y
T
{\displaystyle Y^{T}}
. Then the
G
{\displaystyle {\mathcal {G}}}
-topology on F is compatible with the vector
space structure of F if and only if for every
G
∈
G
{\displaystyle G\in {\mathcal {G}}}
and every f ∈ F, f(G) is bounded in Y.If
Y is locally convex then so is the
G
{\displaystyle {\mathcal {G}}}
-topology on
Y
T
{\displaystyle Y^{T}}
and if
(
p
α
)
{\displaystyle (p_{\alpha })}
is a family of continuous seminorms generating
this topology on Y then the
G
{\displaystyle {\mathcal {G}}}
-topology is induced by the following family
of seminorms:
p
G
,
α
(
f
)
=
sup
x
∈
G
p
α
(
f
(
x
)
)
{\displaystyle p_{G,\alpha }(f)=\sup _{x\in
G}p_{\alpha }(f(x))}
, as G varies over
G
{\displaystyle {\mathcal {G}}}
and
α
{\displaystyle \alpha }
varies over all indices. If Y is Hausdorff
and T is a topological space such that
⋃
G
∈
G
G
{\displaystyle \bigcup _{G\in {\mathcal {G}}}G}
is dense in T then the
G
{\displaystyle {\mathcal {G}}}
-topology on subspace of
Y
T
{\displaystyle Y^{T}}
consisting of all continuous maps is Hausdorff.
If the topological space T is also a topological
vector space, then the condition that
⋃
G
∈
G
G
{\displaystyle \bigcup _{G\in {\mathcal {G}}}G}
be dense in T can be replaced by the weaker
condition that the linear span of this set
be dense in T, in which case we say that this
set is total in T.Let H be a subset of
Y
T
{\displaystyle Y^{T}}
. Then H is bounded in the
G
{\displaystyle {\mathcal {G}}}
-topology if and only if for every
G
∈
G
{\displaystyle G\in {\mathcal {G}}}
,
∪
u
∈
H
u
(
G
)
{\displaystyle \cup _{u\in H}u(G)}
is bounded in Y.
== Spaces of continuous linear maps ==
Throughout this section we will assume that
X and Y are topological vector spaces and
we will let L(X, Y), denote the vector space
of all continuous linear maps from X and Y.
If L(X, Y) is given the
G
{\displaystyle {\mathcal {G}}}
-topology inherited from
Y
X
{\displaystyle Y^{X}}
then this space with this topology is denoted
by
L
G
(
X
,
Y
)
{\displaystyle L_{\mathcal {G}}(X,Y)}
. The
G
{\displaystyle {\mathcal {G}}}
-topology on L(X, Y) is compatible with the
vector space structure of L(X, Y) if and only
if for all
G
∈
G
{\displaystyle G\in {\mathcal {G}}}
and all f ∈ L(X, Y) the set f(G) is bounded
in Y, which we will assume to be the case
for the rest of the article. Note in particular
that this is the case if
G
{\displaystyle {\mathcal {G}}}
consists of (von-Neumann) bounded subsets
of X.
Often,
G
{\displaystyle {\mathcal {G}}}
is required to satisfy the following two axioms:
If
G
{\displaystyle {\mathcal {G}}}
is a bornology on X. which is often the case,
then these two axioms are satisfied.
=== Properties ===
==== Completeness ====
For the following theorems, suppose that X
is a topological vector space and Y is a locally
convex Hausdorff spaces and
G
{\displaystyle {\mathcal {G}}}
is a collection of bounded subsets of X that
satisfies axioms
G
1
{\displaystyle {\mathcal {G}}_{1}}
and
G
2
{\displaystyle {\mathcal {G}}_{2}}
and forms a covering of X.
L
G
(
X
,
Y
)
{\displaystyle L_{\mathcal {G}}(X,Y)}
is complete if
If X is a Mackey space then
L
G
(
X
,
Y
)
{\displaystyle L_{\mathcal {G}}(X,Y)}
is complete if and only if both
X
G
∗
{\displaystyle X_{\mathcal {G}}^{*}}
and Y are complete.
If X is barrelled then
L
G
(
X
,
Y
)
{\displaystyle L_{\mathcal {G}}(X,Y)}
is Hausdorff and quasi-complete, which means
that every closed and bounded set is complete.
==== Boundedness ====
Let X and Y be topological vector space and
H be a subset of L(X, Y). Then the following
are equivalent:
H is bounded in
L
G
(
X
,
Y
)
{\displaystyle L_{\mathcal {G}}(X,Y)}
,
For every
G
∈
G
{\displaystyle G\in {\mathcal {G}}}
,
∪
u
∈
H
u
(
G
)
{\displaystyle \cup _{u\in H}u(G)}
is bounded in Y,
For every neighborhood of 0, V, in Y the set
∩
u
∈
H
u
−
1
(
V
)
{\displaystyle \cap _{u\in H}u^{-1}(V)}
absorbs every
G
∈
G
{\displaystyle G\in {\mathcal {G}}}
.Furthermore,
If X and Y are locally convex Hausdorff space
and if H is bounded in
L
σ
(
X
,
Y
)
{\displaystyle L_{\sigma }(X,Y)}
(i.e. pointwise bounded or simply bounded)
then it is bounded in the topology of uniform
convergence on the convex, balanced, bounded,
complete subsets of X.
If X and Y are locally convex Hausdorff spaces
and if X is quasi-complete (i.e. closed and
bounded subsets are complete), then the bounded
subsets of L(X, Y) are identical for all
G
{\displaystyle {\mathcal {G}}}
-topologies where
G
{\displaystyle {\mathcal {G}}}
is any family of bounded subsets of X covering
X.
If
G
{\displaystyle {\mathcal {G}}}
is any collection of bounded subsets of X
whose union is total in X then every equicontinuous
subset of L(X, Y) is bounded in the
G
{\displaystyle {\mathcal {G}}}
-topology.
=== Examples ===
==== The topology of pointwise convergence
Lσ(X, Y) ====
By letting
G
{\displaystyle {\mathcal {G}}}
be the set of all finite subsets of X, L(X,
Y) will have the weak topology on L(X, Y)
or the topology of pointwise convergence and
L(X, Y) with this topology is denoted by
L
σ
(
X
,
Y
)
{\displaystyle L_{\sigma }(X,Y)}
The weak-topology on L(X, Y) has the following
properties:
The weak-closure of an equicontinuous subset
of L(X, Y) is equicontinuous.
If Y is locally convex, then the convex balanced
hull of an equicontinuous subset of
L
(
X
,
Y
)
{\displaystyle L(X,Y)}
is equicontinuous.
If A ⊆ X is a contable dense subset of a
topological vector space X and if Y is a metrizable
topological vector 
space then
L
σ
(
X
,
Y
)
{\displaystyle L_{\sigma }(X,Y)}
is metrizable.
So in particular, on every equicontinuous
subset of L(X, Y), the topology of pointwise
convergence is metrizable.
Let
Y
X
{\displaystyle Y^{X}}
denote the space of all functions from X into
Y. If
F
(
X
,
Y
)
{\displaystyle F(X,Y)}
is given the topology of pointwise convergence
then space of all linear maps (continuous
or not) X into Y is closed in
Y
X
{\displaystyle Y^{X}}
.
In addition, L(X, Y) is dense in the space
of all linear maps (continuous or not) X into
Y.
==== Compact-convex convergence Lγ(X, Y)
====
By letting
G
{\displaystyle {\mathcal {G}}}
be the set of all compact convex subsets of
X, L(X, Y) will have the topology of compact
convex convergence or the topology of uniform
convergence on compact convex sets L(X, Y)
with this topology is denoted by
L
γ
(
X
,
Y
)
{\displaystyle L_{\gamma }(X,Y)}
.
==== Compact convergence Lc(X, Y) ====
By letting
G
{\displaystyle {\mathcal {G}}}
be the set of all compact subsets of X, L(X,
Y) will have the topology of compact convergence
or the topology of uniform convergence on
compact sets and L(X, Y) with this topology
is denoted by
L
c
(
X
,
Y
)
{\displaystyle L_{c}(X,Y)}
.
The topology of bounded convergence on L(X,
Y) has the following properties:
If X is a Fréchet space or a LF-space and
if Y is a complete locally convex Hausdorff
space then
L
c
(
X
,
Y
)
{\displaystyle L_{c}(X,Y)}
is complete.
On equicontinuous subsets of L(X, Y), the
following topologies coincide:
The topology of pointwise convergence on a
dense subset of X,
The topology of pointwise convergence on X,
The topology of compact convergence.
If X is a Montel space and Y is a topological
vector space, then
L
c
(
X
,
Y
)
{\displaystyle L_{c}(X,Y)}
and
L
b
(
X
,
Y
)
{\displaystyle L_{b}(X,Y)}
have identical topologies.
==== Strong dual topology Lb(X, Y) ====
By letting
G
{\displaystyle {\mathcal {G}}}
be the set of all bounded subsets of X, L(X,
Y) will have the topology of bounded convergence
on X or the topology of uniform convergence
on bounded sets and L(X, Y) with this topology
is denoted by
L
b
(
X
,
Y
)
{\displaystyle L_{b}(X,Y)}
.
The topology of bounded convergence on L(X,
Y) has the following properties:
If X is a bornological space and if Y is a
complete locally convex Hausdorff space then
L
b
(
X
,
Y
)
{\displaystyle L_{b}(X,Y)}
is complete.
If X and Y are both normed spaces then
L
b
(
X
,
Y
)
{\displaystyle L_{b}(X,Y)}
is a normed space with the usual operator
norm.
Every equicontinuous subset of L(X, Y) is
bounded in
L
b
(
X
,
Y
)
{\displaystyle L_{b}(X,Y)}
.
== G-topologies on the continuous dual induced
by X ==
The continuous dual space of a topological
vector space X over the field
F
{\displaystyle {\mathcal {F}}}
(which we will assume to be real or complex
numbers) is the vector space
L
(
X
,
F
)
{\displaystyle L(X,{\mathcal {F}})}
and is denoted by
X
∗
{\displaystyle X^{*}}
and sometimes by
X
′
{\displaystyle X'}
. Given
G
{\displaystyle {\mathcal {G}}}
, a set of subsets of X, we can apply all
of the preceding to this space by using
Y
=
F
{\displaystyle Y={\mathcal {F}}}
and in this case
X
∗
{\displaystyle X^{*}}
with this
G
{\displaystyle {\mathcal {G}}}
-topology is denoted by
X
G
∗
{\displaystyle X_{\mathcal {G}}^{*}}
, so that in particular we have the following
basic properties:
A basis of neighborhoods of 0 for
X
G
∗
{\displaystyle X_{\mathcal {G}}^{*}}
is formed, as
G
{\displaystyle G}
varies over
G
{\displaystyle {\mathcal {G}}}
, by the polar sets
G
∘
:=
{
x
′
∈
X
∗
:
sup
x
∈
G
|
⟨
x
′
,
x
⟩
|
≤
1
}
{\displaystyle G^{\circ }:=\{x'\in X^{*}:\sup
_{x\in G}|\langle x',x\rangle |\leq 1\}}
.
A filter
F
′
{\displaystyle F'}
on
X
∗
{\displaystyle X^{*}}
converges to an element
x
′
∈
X
∗
{\displaystyle x'\in X^{*}}
in the
G
{\displaystyle {\mathcal {G}}}
-topology on
X
∗
{\displaystyle X^{*}}
if
F
′
{\displaystyle F'}
uniformly to
x
′
{\displaystyle x'}
on each
G
∈
G
{\displaystyle G\in {\mathcal {G}}}
.
If G ⊆ X is bounded then
G
∘
{\displaystyle G^{\circ }}
is absorbing, so
G
{\displaystyle {\mathcal {G}}}
usually consists of bounded subsets of X.
X
G
∗
{\displaystyle X_{\mathcal {G}}^{*}}
is locally convex,
If
⋃
G
∈
G
G
{\displaystyle \bigcup _{G\in {\mathcal {G}}}G}
is dense in X then
X
G
∗
{\displaystyle X_{\mathcal {G}}^{*}}
is Hausdorff.
If
⋃
G
∈
G
G
{\displaystyle \bigcup _{G\in {\mathcal {G}}}G}
covers X then the canonical map from X into
(
X
G
∗
)
∗
{\displaystyle (X_{\mathcal {G}}^{*})^{*}}
is well-defined. That is, for all
x
∈
X
{\displaystyle x\in X}
the evaluation functional on
X
∗
{\displaystyle X^{*}}
(i.e.
x
′
∈
X
∗
↦
⟨
x
′
,
x
⟩
{\displaystyle x'\in X^{*}\mapsto \langle
x',x\rangle }
) is continuous on
X
G
∗
{\displaystyle X_{\mathcal {G}}^{*}}
.
If in addition
X
∗
{\displaystyle X^{*}}
separates points on X then the canonical map
of X into
(
X
G
∗
)
∗
{\displaystyle (X_{\mathcal {G}}^{*})^{*}}
is an injection.
Suppose that X and Y are two topological vector
spaces and
u
:
E
→
F
{\displaystyle u:E\to F}
is a continuous linear map. Suppose that
G
{\displaystyle {\mathcal {G}}}
and
H
{\displaystyle {\mathcal {H}}}
are collections of bounded subsets of X and
Y, respectively, that both satisfy axioms
G
1
{\displaystyle {\mathcal {G}}_{1}}
and
G
2
{\displaystyle {\mathcal {G}}_{2}}
. Then
u
{\displaystyle u}
's transpose,
t
u
:
Y
H
∗
→
X
G
∗
{\displaystyle {}^{t}u:Y_{\mathcal {H}}^{*}\to
X_{\mathcal {G}}^{*}}
is continuous if for every
G
∈
G
{\displaystyle G\in {\mathcal {G}}}
there is a
H
∈
H
{\displaystyle H\in {\mathcal {H}}}
such that u(G) ⊆ H.In particular, the transpose
of
u
{\displaystyle u}
is continuous if
X
∗
{\displaystyle X^{*}}
carries the
σ
(
X
∗
,
X
)
{\displaystyle \sigma (X^{*},X)}
(respectively,
γ
(
X
∗
,
X
)
{\displaystyle \gamma (X^{*},X)}
,
c
(
X
∗
,
X
)
{\displaystyle c(X^{*},X)}
,
b
(
X
∗
,
X
)
{\displaystyle b(X^{*},X)}
) topology and
Y
∗
{\displaystyle Y^{*}}
carry any topology stronger than the
σ
(
Y
∗
,
Y
)
{\displaystyle \sigma (Y^{*},Y)}
topology (respectively,
γ
(
Y
∗
,
Y
)
{\displaystyle \gamma (Y^{*},Y)}
,
c
(
Y
∗
,
Y
)
{\displaystyle c(Y^{*},Y)}
,
b
(
Y
∗
,
Y
)
{\displaystyle b(Y^{*},Y)}
).
If X is a locally convex Hausdorff topological
vector space over the field
F
{\displaystyle {\mathcal {F}}}
and
G
{\displaystyle {\mathcal {G}}}
is a collection of bounded subsets of X that
satisfies axioms
G
1
{\displaystyle {\mathcal {G}}_{1}}
and
G
2
{\displaystyle {\mathcal {G}}_{2}}
then the bilinear map
X
×
X
G
∗
→
F
{\displaystyle X\times X_{\mathcal {G}}^{*}\to
{\mathcal {F}}}
defined by
(
x
,
x
′
)
↦
⟨
x
′
,
x
⟩
=
x
′
(
x
)
{\displaystyle (x,x')\mapsto \langle x',x\rangle
=x'(x)}
is continuous if and only if X is normable
and the
G
{\displaystyle {\mathcal {G}}}
-topology on
X
∗
{\displaystyle X^{*}}
is the strong dual topology
b
(
X
∗
,
X
)
{\displaystyle b(X^{*},X)}
.
Suppose that X is a Fréchet space and
G
{\displaystyle {\mathcal {G}}}
is a collection of bounded subsets of X that
satisfies axioms
G
1
{\displaystyle {\mathcal {G}}_{1}}
and
G
2
{\displaystyle {\mathcal {G}}_{2}}
. If
G
{\displaystyle {\mathcal {G}}}
contains all compact subsets of X then
X
G
∗
{\displaystyle X_{\mathcal {G}}^{*}}
is complete.
=== Examples ===
==== The weak topology σ(X*, X) or the weak*
topology ====
By letting
G
{\displaystyle {\mathcal {G}}}
be the set of all finite subsets of X,
X
∗
{\displaystyle X^{*}}
will have the weak topology on
X
∗
{\displaystyle X^{*}}
more commonly known as the weak* topology
or the topology of pointwise convergence,
which is denoted by
σ
(
X
∗
,
X
)
{\displaystyle \sigma (X^{*},X)}
and
X
∗
{\displaystyle X^{*}}
with this topology is denoted by
X
σ
∗
{\displaystyle X_{\sigma }^{*}}
or by
X
σ
(
X
∗
,
X
)
∗
{\displaystyle X_{\sigma (X^{*},X)}^{*}}
if there may be ambiguity.
The
σ
(
X
∗
,
X
)
{\displaystyle \sigma (X^{*},X)}
topology has the following properties:
Theorem (S. Banach): Suppose that X and Y
are Fréchet spaces or that they are duals
of reflexive Fréchet spaces and that
u
:
X
→
Y
{\displaystyle u:X\to Y}
is a continuous linear map. Then
u
{\displaystyle u}
is surjective if and only if the transpose
of
u
{\displaystyle u}
,
t
u
:
Y
∗
→
X
∗
{\displaystyle {}^{t}u:Y^{*}\to X^{*}}
, is one-to-one and the range of
t
u
{\displaystyle {}^{t}u}
is weakly closed in
X
σ
(
X
∗
,
X
)
∗
{\displaystyle X_{\sigma (X^{*},X)}^{*}}
.
Suppose that X and Y are Fréchet spaces,
Z
{\displaystyle Z}
is a Hausdorff locally convex space and that
u
:
X
σ
∗
×
Y
σ
∗
→
Z
σ
∗
{\displaystyle u:X_{\sigma }^{*}\times Y_{\sigma
}^{*}\to Z_{\sigma }^{*}}
is a separately-continuous bilinear map. Then
u
:
X
b
∗
×
Y
b
∗
→
Z
b
∗
{\displaystyle u:X_{b}^{*}\times Y_{b}^{*}\to
Z_{b}^{*}}
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.
X
σ
(
X
∗
,
X
)
∗
{\displaystyle X_{\sigma (X^{*},X)}^{*}}
is normable if and only if X is finite-dimensional.
When X is infinite-dimensional the
σ
(
X
∗
,
X
)
{\displaystyle \sigma (X^{*},X)}
topology on
X
∗
{\displaystyle X^{*}}
is strictly less fine than the strong dual
topology
b
(
X
∗
,
X
)
{\displaystyle b(X^{*},X)}
.
The
σ
(
X
∗
,
X
)
{\displaystyle \sigma (X^{*},X)}
-closure of the convex balanced hull of an
equicontinuous subset of
X
∗
{\displaystyle X^{*}}
is equicontinuous and
σ
(
X
∗
,
X
)
{\displaystyle \sigma (X^{*},X)}
-compact.
Suppose that X is a locally convex Hausdorff
space and that
X
^
{\displaystyle {\hat {X}}}
is its completion. If
X
≠
X
^
{\displaystyle X\neq {\hat {X}}}
then
σ
(
X
∗
,
X
^
)
{\displaystyle \sigma (X^{*},{\hat {X}})}
is strictly finer than
σ
(
X
∗
,
X
)
{\displaystyle \sigma (X^{*},X)}
.
Any equicontinuous subset in the dual of a
separable Hausdorff locally convex vector
space is metrizable in the
σ
(
X
∗
,
X
)
{\displaystyle \sigma (X^{*},X)}
topology.
==== Compact-convex convergence γ(X*, X)
====
By letting
G
{\displaystyle {\mathcal {G}}}
be the set of all compact convex subsets of
X,
X
∗
{\displaystyle X^{*}}
will have the topology of compact convex convergence
or the topology of uniform convergence on
compact convex sets, which is denoted by
γ
(
X
∗
,
X
)
{\displaystyle \gamma (X^{*},X)}
and
X
∗
{\displaystyle X^{*}}
with this topology is denoted by
X
γ
∗
{\displaystyle X_{\gamma }^{*}}
or by
X
γ
(
X
∗
,
X
)
∗
{\displaystyle X_{\gamma (X^{*},X)}^{*}}
.
If X is a Fréchet space then the topologies
γ
(
X
∗
,
X
)
=
c
(
X
∗
,
X
)
{\displaystyle \gamma (X^{*},X)=c(X^{*},X)}
.
==== Compact convergence c(X*, X) ====
By letting
G
{\displaystyle {\mathcal {G}}}
be the set of all compact subsets of X,
X
∗
{\displaystyle X^{*}}
will have the topology of compact convergence
or the topology of uniform convergence on
compact sets, which is denoted by
c
(
X
∗
,
X
)
{\displaystyle c(X^{*},X)}
and
X
∗
{\displaystyle X^{*}}
with this topology is denoted by
X
c
∗
{\displaystyle X_{c}^{*}}
or by
X
c
(
X
∗
,
X
)
∗
{\displaystyle X_{c(X^{*},X)}^{*}}
.
If X is a Fréchet space or a LF-space then
c
(
X
∗
,
X
)
{\displaystyle c(X^{*},X)}
is complete.
Suppose that X is a metrizable topological
vector space and that
W
′
⊆
X
∗
{\displaystyle W'\subseteq X^{*}}
. If the intersection of
W
′
{\displaystyle W'}
with every equicontinuous subset of
X
∗
{\displaystyle X^{*}}
is weakly-open, then
W
′
{\displaystyle W'}
is open in
c
(
X
∗
,
X
)
{\displaystyle c(X^{*},X)}
.
==== Precompact convergence ====
By letting
G
{\displaystyle {\mathcal {G}}}
be the set of all precompact subsets of X,
X
∗
{\displaystyle X^{*}}
will have the topology of precompact convergence
or the topology of uniform convergence on
precompact sets.
Alaoglu–Bourbaki Theorem: An equicontinuous
subset K of
X
∗
{\displaystyle X^{*}}
has compact closure in the topology of uniform
convergence on precompact sets. Furthermore,
this topology on K coincides with the
σ
(
X
∗
,
X
)
{\displaystyle \sigma (X^{*},X)}
topology.
==== Mackey topology τ(X*, X) ====
By letting
G
{\displaystyle {\mathcal {G}}}
be the set of all convex balanced weakly compact
subsets of X,
X
∗
{\displaystyle X^{*}}
will have the Mackey topology on
X
∗
{\displaystyle X^{*}}
or the topology of uniform convergence on
convex balanced weakly compact sets, which
is denoted by
τ
(
X
∗
,
X
)
{\displaystyle \tau (X^{*},X)}
and
X
∗
{\displaystyle X^{*}}
with this topology is denoted by
X
τ
(
X
∗
,
X
)
∗
{\displaystyle X_{\tau (X^{*},X)}^{*}}
.
==== Strong dual topology b(X*, X) ====
By letting
G
{\displaystyle {\mathcal {G}}}
be the set of all bounded subsets of X,
X
∗
{\displaystyle X^{*}}
will have the topology of bounded convergence
on X or the topology of uniform convergence
on bounded sets or the strong dual topology
on
X
∗
{\displaystyle X^{*}}
, which is denoted by
b
(
X
∗
,
X
)
{\displaystyle b(X^{*},X)}
and
X
∗
{\displaystyle X^{*}}
with this topology is denoted by
X
b
∗
{\displaystyle X_{b}^{*}}
or by
X
b
(
X
∗
,
X
)
∗
{\displaystyle X_{b(X^{*},X)}^{*}}
. Due to its importance, the continuous dual
space of
X
b
∗
{\displaystyle X_{b}^{*}}
, which is commonly denoted by
X
∗
∗
{\displaystyle X^{**}}
so that
(
X
b
∗
)
∗
=
X
∗
∗
{\displaystyle (X_{b}^{*})^{*}=X^{**}}
.
The
b
(
X
∗
,
X
)
{\displaystyle b(X^{*},X)}
topology has the following properties:
If X is locally convex, then this topology
is finer than all other
G
{\displaystyle {\mathcal {G}}}
-topologies on
X
∗
{\displaystyle X^{*}}
when considering only
G
{\displaystyle {\mathcal {G}}}
's whose sets are subsets of X.
If X is a bornological space (ex: metrizable
or LF-space) then
X
b
(
X
∗
,
X
)
∗
{\displaystyle X_{b(X^{*},X)}^{*}}
is complete.
If X is a normed space then the strong dual
topology on
X
∗
{\displaystyle X^{*}}
may be defined by the norm
‖
x
′
‖
=
sup
x
∈
X
,
,
‖
x
‖
=
1
|
⟨
x
′
,
x
⟩
|
{\displaystyle \|x'\|=\sup _{x\in X,,\|x\|=1}|\langle
x',x\rangle |}
, where
x
′
∈
X
∗
{\displaystyle x'\in X^{*}}
.
If X is a LF-space that is the inductive limit
of the sequence of space
X
k
{\displaystyle X_{k}}
(for
k
=
0
,
1
…
{\displaystyle k=0,1\dots }
) then
X
b
(
X
∗
,
X
)
∗
{\displaystyle X_{b(X^{*},X)}^{*}}
is a Fréchet space if and only if all
X
k
{\displaystyle X_{k}}
are normable.
If X is a Montel space then
X
b
(
X
∗
,
X
)
∗
{\displaystyle X_{b(X^{*},X)}^{*}}
has the Heine–Broel property (i.e. every
closed and bounded subset of
X
b
(
X
∗
,
X
)
∗
{\displaystyle X_{b(X^{*},X)}^{*}}
is compact in
X
b
(
X
∗
,
X
)
∗
{\displaystyle X_{b(X^{*},X)}^{*}}
)
On bounded subsets of
X
b
(
X
∗
,
X
)
∗
{\displaystyle X_{b(X^{*},X)}^{*}}
, the strong and weak topologies coincide
(and hence so do all other topologies finer
than
σ
(
X
∗
,
X
)
{\displaystyle \sigma (X^{*},X)}
and coarser than
b
(
X
∗
,
X
)
{\displaystyle b(X^{*},X)}
).
Every weakly convergent sequence in
X
∗
{\displaystyle X^{*}}
is strongly convergent.
==== Mackey topology τ(X*, X**) ====
By letting
G
″
{\displaystyle {\mathcal {G''}}}
be the set of all convex balanced weakly compact
subsets of
X
∗
∗
=
(
X
b
∗
)
∗
{\displaystyle X^{**}=(X_{b}^{*})^{*}}
,
X
∗
{\displaystyle X^{*}}
will have the Mackey topology on
X
∗
{\displaystyle X^{*}}
induced by
X
∗
∗
{\displaystyle X^{**}}
' or the topology of uniform convergence on
convex balanced weakly compact subsets of
X
∗
∗
{\displaystyle X^{**}}
, which is denoted by
τ
(
X
∗
,
X
∗
∗
)
{\displaystyle \tau (X^{*},X^{**})}
and
X
∗
{\displaystyle X^{*}}
with this topology is denoted by
X
τ
(
X
∗
,
X
∗
∗
)
∗
{\displaystyle X_{\tau (X^{*},X^{**})}^{*}}
.
This topology is finer than
b
(
X
∗
,
X
)
{\displaystyle b(X^{*},X)}
and hence finer than
τ
(
X
∗
,
X
)
{\displaystyle \tau (X^{*},X)}
.
==== Other examples ====
Other
G
{\displaystyle {\mathcal {G}}}
-topologies on
X
∗
{\displaystyle X^{*}}
include
The topology of uniform convergence on convex
balanced complete bounded subsets of X.
The topology of uniform convergence on convex
balanced infracomplete bounded subsets of
X.
== G-topologies on X induced by the continuous
dual ==
There is a canonical map from X into
(
X
σ
∗
)
∗
{\displaystyle (X_{\sigma }^{*})^{*}}
which maps an element
x
∈
X
{\displaystyle x\in X}
to the following map:
x
′
∈
X
∗
↦
⟨
x
′
,
x
⟩
{\displaystyle x'\in X^{*}\mapsto \langle
x',x\rangle }
. By using this canonical map we can identify
X as being contained in the continuous dual
of
X
σ
∗
{\displaystyle X_{\sigma }^{*}}
i.e. contained in
(
X
σ
∗
)
∗
{\displaystyle (X_{\sigma }^{*})^{*}}
. In fact, this canonical map is onto, which
means that
X
=
(
X
σ
∗
)
∗
{\displaystyle X=(X_{\sigma }^{*})^{*}}
so that we can through this canonical isomorphism
think of X as the continuous dual space of
X
σ
∗
{\displaystyle X_{\sigma }^{*}}
. Note that it is a common convention that
if an equal sign appears between two sets
which are clearly not equal, then the equality
really means that the sets are isomorphic
through some canonical map.
Since we are now regarding X as the continuous
dual space of
X
σ
∗
{\displaystyle X_{\sigma }^{*}}
, we can look at sets of subsets of
X
σ
∗
{\displaystyle X_{\sigma }^{*}}
, say
G
′
{\displaystyle {\mathcal {G'}}}
and construct a dual space topology on the
dual of
X
σ
∗
{\displaystyle X_{\sigma }^{*}}
, which is X. * A basis of neighborhoods of
0 for
X
G
′
{\displaystyle X_{\mathcal {G'}}}
is formed by the Polar sets
G
′
∘
:=
{
x
∈
X
:
sup
x
′
∈
G
′
|
⟨
x
′
,
x
⟩
|
≤
1
}
{\displaystyle G'^{\circ }:=\{x\in X:\sup
_{x'\in G'}|\langle x',x\rangle |\leq 1\}}
as
G
′
{\displaystyle G'}
varies over
G
′
{\displaystyle {\mathcal {G'}}}
.
=== Examples ===
==== The weak topology σ(X, X*) ====
By letting
G
′
{\displaystyle {\mathcal {G'}}}
be the set of all finite subsets of
X
′
{\displaystyle X'}
, X will have the weak topology or the topology
of pointwise convergence on
X
∗
{\displaystyle X^{*}}
, which is denoted 
by
σ
(
X
,
X
∗
)
{\displaystyle \sigma (X,X^{*})}
and X with this topology is denoted by
X
σ
{\displaystyle X_{\sigma }}
or by
X
σ
(
X
,
X
∗
)
{\displaystyle X_{\sigma (X,X^{*})}}
if there may be ambiguity.
Suppose that X and Y are Hausdorff locally
convex spaces with X metrizable and that
u
:
X
→
Y
{\displaystyle u:X\to Y}
is a linear map. Then
u
:
X
→
Y
{\displaystyle u:X\to Y}
is continuous if and only if
u
:
σ
(
X
,
X
∗
)
→
σ
(
Y
,
Y
∗
)
{\displaystyle u:\sigma (X,X^{*})\to \sigma
(Y,Y^{*})}
is continuous. That is,
u
:
X
→
Y
{\displaystyle u:X\to Y}
is continuous when X and Y carry their given
topologies if and only if
u
{\displaystyle u}
is continuous when X and Y carry their weak
topologies.
==== Convergence on equicontinuous sets ε(X,
X*) ====
By letting
G
′
{\displaystyle {\mathcal {G'}}}
be the set of all equicontinuous subsets
X
∗
{\displaystyle X^{*}}
, X will have the topology of uniform convergence
on equicontinuous subsets of
X
∗
{\displaystyle X^{*}}
, which is denoted by
ϵ
(
X
,
X
∗
)
{\displaystyle \epsilon (X,X^{*})}
and X with this topology is denoted by
X
ϵ
{\displaystyle X_{\epsilon }}
or by
X
ϵ
(
X
,
X
∗
)
{\displaystyle X_{\epsilon (X,X^{*})}}
.
If
G
′
{\displaystyle {\mathcal {G'}}}
was the set of all convex balanced weakly
compact equicontinuous subsets of
X
∗
{\displaystyle X^{*}}
, then the same topology would have been induced.
If X is locally convex and Hausdorff then
X's given topology (i.e. the topology that
X started with) is exactly
ϵ
(
X
,
X
∗
)
{\displaystyle \epsilon (X,X^{*})}
.
==== Mackey topology τ(X, X*) ====
By letting
G
′
{\displaystyle {\mathcal {G'}}}
be the set of all convex balanced weakly compact
subsets of
X
∗
{\displaystyle X^{*}}
, X will have the Mackey topology on X or
the topology of uniform convergence on convex
balanced weakly compact subsets of
X
∗
{\displaystyle X^{*}}
, which is denoted by
τ
(
X
,
X
∗
)
{\displaystyle \tau (X,X^{*})}
and X with this topology is denoted by
X
τ
{\displaystyle X_{\tau }}
or by
X
τ
(
X
,
X
∗
)
{\displaystyle X_{\tau (X,X^{*})}}
.
Suppose that X is a locally convex Hausdorff
space. If X is metrizable or barrelled then
the initial topology of X is identical to
the Mackey topology
τ
(
X
,
X
∗
)
{\displaystyle \tau (X,X^{*})}
.
==== Bounded convergence b(X, X*) ====
By letting
G
{\displaystyle {\mathcal {G}}}
be the set of all bounded subsets of X,
X
∗
{\displaystyle X^{*}}
will have the topology of bounded convergence
or the topology of uniform convergence on
bounded sets, which is denoted by
b
(
X
,
X
∗
)
{\displaystyle b(X,X^{*})}
and
X
∗
{\displaystyle X^{*}}
with this topology is denoted by
X
b
∗
{\displaystyle X_{b}^{*}}
or by
X
b
(
X
,
X
∗
)
∗
{\displaystyle X_{b(X,X^{*})}^{*}}
.
=== The Mackey–Arens theorem ===
Let X 
be a vector space and let Y be a vector subspace
of the algebraic dual of X that separates
points on X. Any locally convex Hausdorff
topological vector space (TVS) topology on
X with the property that when X is equipped
with this topology has Y as its continuous
dual space is said to be compatible with duality
between X and Y. If we give X the weak topology
σ
(
X
,
Y
)
{\displaystyle \sigma (X,Y)}
then
X
σ
(
X
,
Y
)
{\displaystyle X_{\sigma (X,Y)}}
is a Hausdorff locally convex topological
vector space (TVS) and
σ
(
X
,
Y
)
{\displaystyle \sigma (X,Y)}
is compatible with duality between X and Y
(i.e.
X
σ
(
X
,
Y
)
∗
=
(
X
σ
(
X
,
Y
)
)
∗
=
Y
{\displaystyle X_{\sigma (X,Y)}^{*}=(X_{\sigma
(X,Y)})^{*}=Y}
). We can now ask the question: what are all
of the locally convex Hausdorff TVS topologies
that we can place on X that are compatible
with duality between X and Y? The answer to
this question is called the Mackey–Arens
theorem:
Theorem. Let X be a vector space and let
T
{\displaystyle {\mathcal {T}}}
be a locally convex Hausdorff topological
vector space topology on X. Let
X
∗
{\displaystyle X^{*}}
denote the continuous dual space of X and
let
X
T
{\displaystyle X_{\mathcal {T}}}
denote X with the topology
T
{\displaystyle {\mathcal {T}}}
. Then the following are equivalent:
And furthermore,
== G-H-topologies on spaces of bilinear maps
==
We will let
B
(
X
,
Y
;
Z
)
{\displaystyle {\mathcal {B}}(X,Y;Z)}
denote the space of separately continuous
bilinear maps and
B
(
X
,
Y
;
Z
)
{\displaystyle B(X,Y;Z)}
denote its subspace the space of continuous
bilinear maps, where
X
,
Y
{\displaystyle X,Y}
and
Z
{\displaystyle Z}
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 L(X, Y) we can place a topology on
B
(
X
,
Y
;
Z
)
{\displaystyle {\mathcal {B}}(X,Y;Z)}
and
B
(
X
,
Y
;
Z
)
{\displaystyle B(X,Y;Z)}
.
Let
G
{\displaystyle {\mathcal {G}}}
be a set of subsets of X,
H
{\displaystyle {\mathcal {H}}}
be a set of subsets of Y. Let
G
×
H
{\displaystyle {\mathcal {G}}\times {\mathcal
{H}}}
denote the collection of all sets G × H where
G
∈
G
{\displaystyle G\in {\mathcal {G}}}
,
H
∈
H
{\displaystyle H\in {\mathcal {H}}}
. We can place on
Z
X
×
Y
{\displaystyle Z^{X\times Y}}
the
G
×
H
{\displaystyle {\mathcal {G}}\times {\mathcal
{H}}}
-topology, and consequently on any of its
subsets, in particular on
B
(
X
,
Y
;
Z
)
{\displaystyle B(X,Y;Z)}
and on
B
(
X
,
Y
;
Z
)
{\displaystyle {\mathcal {B}}(X,Y;Z)}
. This topology is known as the
G
−
H
{\displaystyle {\mathcal {G}}-{\mathcal {H}}}
-topology or as the topology of uniform convergence
on the products
G
×
H
{\displaystyle G\times H}
of
G
×
H
{\displaystyle {\mathcal {G}}\times {\mathcal
{H}}}
.
However, as before, this topology is not necessarily
compatible with the vector space structure
of
B
(
X
,
Y
;
Z
)
{\displaystyle {\mathcal {B}}(X,Y;Z)}
or of
B
(
X
,
Y
;
Z
)
{\displaystyle B(X,Y;Z)}
without the additional requirement that for
all bilinear maps,
b
{\displaystyle b}
in 
this space (that is, in
B
(
X
,
Y
;
Z
)
{\displaystyle {\mathcal {B}}(X,Y;Z)}
or in
B
(
X
,
Y
;
Z
)
{\displaystyle B(X,Y;Z)}
) and for all
G
∈
G
{\displaystyle G\in {\mathcal {G}}}
and
H
∈
H
{\displaystyle H\in {\mathcal {H}}}
the set
b
(
G
,
H
)
{\displaystyle b(G,H)}
is bounded in X. If both
G
{\displaystyle {\mathcal {G}}}
and
H
{\displaystyle {\mathcal {H}}}
consist of bounded sets then this requirement
is automatically satisfied if we are topologizing
B
(
X
,
Y
;
Z
)
{\displaystyle B(X,Y;Z)}
but this may not be the case if we are trying
to topologize
B
(
X
,
Y
;
Z
)
{\displaystyle {\mathcal {B}}(X,Y;Z)}
. The
G
{\displaystyle {\mathcal {G}}}
-
H
{\displaystyle {\mathcal {H}}}
-topology on
B
(
X
,
Y
;
Z
)
{\displaystyle {\mathcal {B}}(X,Y;Z)}
will be compatible with the vector space structure
of
B
(
X
,
Y
;
Z
)
{\displaystyle {\mathcal {B}}(X,Y;Z)}
if both
G
{\displaystyle {\mathcal {G}}}
and
H
{\displaystyle {\mathcal {H}}}
consists of bounded sets and any of the following
conditions hold:
X and Y are barrelled spaces and
Z
{\displaystyle Z}
is locally convex.
X is a F-space, Y is metrizable, and
Z
{\displaystyle Z}
is Hausdorff, in which case
B
(
X
,
Y
;
Z
)
=
B
(
X
,
Y
;
Z
)
{\displaystyle {\mathcal {B}}(X,Y;Z)=B(X,Y;Z)}
,.
X
,
Y
{\displaystyle X,Y}
, and
Z
{\displaystyle Z}
are the strong duals of reflexive Fréchet
spaces.
X is normed and Y and
Z
{\displaystyle Z}
the strong duals of reflexive Fréchet spaces.
=== The ε-topology ===
Suppose that
X
,
Y
{\displaystyle X,Y}
, and
Z
{\displaystyle Z}
are locally convex spaces and let
G
{\displaystyle {\mathcal {G}}}
' and
H
{\displaystyle {\mathcal {H}}}
' be the collections of equicontinuous subsets
of
X
∗
{\displaystyle X^{*}}
and
Y
∗
{\displaystyle Y^{*}}
, respectively. Then the
G
{\displaystyle {\mathcal {G}}}
'-
H
{\displaystyle {\mathcal {H}}}
'-topology on
B
(
X
b
(
X
∗
,
X
)
∗
,
Y
b
(
X
∗
,
X
)
∗
;
Z
)
{\displaystyle {\mathcal {B}}(X_{b(X^{*},X)}^{*},Y_{b(X^{*},X)}^{*};Z)}
will be a topological vector space topology.
This topology is called the ε-topology and
B
(
X
b
(
X
∗
,
X
)
∗
,
Y
b
(
X
∗
,
X
)
;
Z
)
{\displaystyle {\mathcal {B}}(X_{b(X^{*},X)}^{*},Y_{b(X^{*},X)};Z)}
with this topology it is denoted by
B
ϵ
(
X
b
(
X
∗
,
X
)
∗
,
Y
b
(
X
∗
,
X
)
∗
;
Z
)
{\displaystyle {\mathcal {B}}_{\epsilon }(X_{b(X^{*},X)}^{*},Y_{b(X^{*},X)}^{*};Z)}
or simply by
B
ϵ
(
X
b
∗
,
Y
b
∗
;
Z
)
{\displaystyle {\mathcal {B}}_{\epsilon }(X_{b}^{*},Y_{b}^{*};Z)}
.
Part of the importance of this vector space
and this topology is that it contains many
subspace, such as
B
(
X
σ
(
X
∗
,
X
)
∗
,
Y
σ
(
X
∗
,
X
)
∗
;
Z
)
{\displaystyle {\mathcal {B}}(X_{\sigma (X^{*},X)}^{*},Y_{\sigma
(X^{*},X)}^{*};Z)}
, which we denote by
B
(
X
σ
∗
,
Y
σ
∗
;
Z
)
{\displaystyle {\mathcal {B}}(X_{\sigma }^{*},Y_{\sigma
}^{*};Z)}
. When this subspace is given the subspace
topology of
B
ϵ
{\displaystyle {\mathcal {B}}_{\epsilon }}
(
X
b
∗
,
Y
b
∗
;
Z
)
{\displaystyle (X_{b}^{*},Y_{b}^{*};Z)}
it is denoted by
B
ϵ
(
X
σ
∗
,
Y
σ
∗
;
Z
)
{\displaystyle {\mathcal {B}}_{\epsilon }(X_{\sigma
}^{*},Y_{\sigma }^{*};Z)}
.
In the instance where Z is the field of these
vector spaces
B
(
X
σ
∗
,
Y
σ
∗
)
{\displaystyle {\mathcal {B}}(X_{\sigma }^{*},Y_{\sigma
}^{*})}
is a tensor product of X and Y. In fact, if
X and Y are locally convex Hausdorff spaces
then
B
(
X
σ
∗
,
Y
σ
∗
)
{\displaystyle {\mathcal {B}}(X_{\sigma }^{*},Y_{\sigma
}^{*})}
is vector space isomorphic to
L
(
X
σ
(
X
∗
,
X
)
∗
,
Y
σ
(
Y
∗
,
Y
)
)
{\displaystyle L(X_{\sigma (X^{*},X)}^{*},Y_{\sigma
(Y^{*},Y)})}
, which is in turn equal to
L
(
X
τ
(
X
∗
,
X
)
∗
,
Y
)
{\displaystyle L(X_{\tau (X^{*},X)}^{*},Y)}
.
These spaces have the following properties:
If X and Y are locally convex Hausdorff spaces
then
B
ϵ
{\displaystyle {\mathcal {B}}_{\epsilon }}
(
X
σ
∗
,
Y
σ
∗
)
{\displaystyle (X_{\sigma }^{*},Y_{\sigma
}^{*})}
is complete if and only if both X and Y are
complete.
If X and Y are both normed (or both Banach)
then so is
B
ϵ
{\displaystyle {\mathcal {B}}_{\epsilon }}
(
X
σ
∗
,
Y
σ
∗
)
{\displaystyle (X_{\sigma }^{*},Y_{\sigma
}^{*})}
== See also ==
Bornological space
Bounded linear operator
Operator norm
Uniform convergence
Uniform space
Polar topology
== Notes
