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A dual pair of vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s0906101.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s0906102.png" /> is a pair of vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s0906103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s0906104.png" /> together with a non-degenerate bilinear form over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s0906105.png" />,
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s0906106.png" /></td> </tr></table>
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I.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s0906107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s0906108.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s0906109.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061010.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061011.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061012.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061013.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061014.png" />.
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A dual pair of vector spaces  $  ( L, M) $
 +
over a field  $  k $
 +
is a pair of vector spaces  $  L $,
 +
$  M $
 +
together with a non-degenerate bilinear form over  $  k $,
  
The [[Weak topology|weak topology]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061015.png" /> defined by the dual pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061016.png" /> (given a topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061017.png" />) is the weakest topology such that all the functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061019.png" />, are continuous. More precisely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061020.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061021.png" /> with the usual topology, this defines the weak topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061022.png" /> (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061023.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061024.png" /> is an arbitrary field with the discrete topology, this defines the so-called linear weak topology.
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$$
 +
\phi : L \times M  \rightarrow  k.
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061025.png" /> be a collection of bounded subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061026.png" /> (for the weak topology, i.e. every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061027.png" /> is weakly bounded, meaning that for every open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061028.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061029.png" /> in the weak topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061030.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061031.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061032.png" />). The topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061033.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061034.png" /> is defined by the system of semi-norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061036.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061037.png" /> (cf. [[Semi-norm|Semi-norm]]). This topology is locally convex if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061038.png" /> is a total set, i.e. it generates (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061039.png" /> as a vector space) all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061040.png" />. The topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061041.png" /> is called the topology of uniform convergence on the sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061043.png" />.
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I.e. $  \phi ( a _ {1} l _ {1} + a _ {2} l _ {2} , m)= a _ {1} \phi ( l _ {1} , m)+ a _ {2} \phi ( l _ {2} , m) $,  
 +
$  \phi ( l, b _ {1} m _ {1} + b _ {2} m _ {2} ) = b _ {1} \phi ( l, m _ {1} )+ b _ {2} \phi ( l, m _ {2} ) $;
 +
$  \phi ( l, m)= 0 $
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for all $  m \in M $
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implies  $  l= 0 $;
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$  \phi ( l, m)= 0 $
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for all  $  l \in L $
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implies  $  m= 0 $.
  
The finest topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061044.png" /> which can be defined in terms of the dual pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061045.png" /> is the topology of uniform convergence on weakly bounded subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061046.png" />. This is the topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061047.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061048.png" /> is the collection of all weakly bounded subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061049.png" />, and it is called the strong topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090610/s09061050.png" />, for brevity.
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The [[Weak topology|weak topology]] on  $  L $
 +
defined by the dual pair  $  ( L, M) $(
 +
given a topology on $  k $)
 +
is the weakest topology such that all the functionals  $  \psi _ {m} :  L \rightarrow k $,
 +
$  \psi _ {m} ( l) = \phi ( l, m) $,
 +
are continuous. More precisely, if  $  k = \mathbf R $
 +
or  $  \mathbf C $
 +
with the usual topology, this defines the weak topology on  $  L $(
 +
and  $  M $).
 +
If  $  k $
 +
is an arbitrary field with the discrete topology, this defines the so-called linear weak topology.
 +
 
 +
Let  $  \mathfrak M $
 +
be a collection of bounded subsets of  $  L $(
 +
for the weak topology, i.e. every  $  A \in \mathfrak M $
 +
is weakly bounded, meaning that for every open subset  $  U $
 +
of  $  0 $
 +
in the weak topology on  $  L $
 +
there is a  $  \rho > 0 $
 +
such that  $  \rho A \subset  U $).
 +
The topology  $  \tau _ {\mathfrak M }  $
 +
on  $  M $
 +
is defined by the system of semi-norms  $  \{ \rho _ {A} \} $,
 +
$  A \in \mathfrak M $,
 +
where  $  \rho _ {A} ( x) = \sup _ {m \in A }  | \phi ( m, x) | $(
 +
cf. [[Semi-norm|Semi-norm]]). This topology is locally convex if and only if  $  \cup \mathfrak M $
 +
is a total set, i.e. it generates (in  $  L $
 +
as a vector space) all of  $  L $.  
 +
The topology  $  \tau _ {\mathfrak M }  $
 +
is called the topology of uniform convergence on the sets of  $  \mathfrak M $.
 +
 
 +
The finest topology on  $  M $
 +
which can be defined in terms of the dual pairs $  ( L, M) $
 +
is the topology of uniform convergence on weakly bounded subsets of $  L $.  
 +
This is the topology $  \tau _ {\mathfrak M }  $
 +
where $  \mathfrak M $
 +
is the collection of all weakly bounded subsets of $  L $,  
 +
and it is called the strong topology on $  M $,  
 +
for brevity.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Köthe,  "Topological vector spaces" , '''1''' , Springer  (1969)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Köthe,  "Topological vector spaces" , '''1''' , Springer  (1969)</TD></TR></table>

Latest revision as of 08:24, 6 June 2020


A dual pair of vector spaces $ ( L, M) $ over a field $ k $ is a pair of vector spaces $ L $, $ M $ together with a non-degenerate bilinear form over $ k $,

$$ \phi : L \times M \rightarrow k. $$

I.e. $ \phi ( a _ {1} l _ {1} + a _ {2} l _ {2} , m)= a _ {1} \phi ( l _ {1} , m)+ a _ {2} \phi ( l _ {2} , m) $, $ \phi ( l, b _ {1} m _ {1} + b _ {2} m _ {2} ) = b _ {1} \phi ( l, m _ {1} )+ b _ {2} \phi ( l, m _ {2} ) $; $ \phi ( l, m)= 0 $ for all $ m \in M $ implies $ l= 0 $; $ \phi ( l, m)= 0 $ for all $ l \in L $ implies $ m= 0 $.

The weak topology on $ L $ defined by the dual pair $ ( L, M) $( given a topology on $ k $) is the weakest topology such that all the functionals $ \psi _ {m} : L \rightarrow k $, $ \psi _ {m} ( l) = \phi ( l, m) $, are continuous. More precisely, if $ k = \mathbf R $ or $ \mathbf C $ with the usual topology, this defines the weak topology on $ L $( and $ M $). If $ k $ is an arbitrary field with the discrete topology, this defines the so-called linear weak topology.

Let $ \mathfrak M $ be a collection of bounded subsets of $ L $( for the weak topology, i.e. every $ A \in \mathfrak M $ is weakly bounded, meaning that for every open subset $ U $ of $ 0 $ in the weak topology on $ L $ there is a $ \rho > 0 $ such that $ \rho A \subset U $). The topology $ \tau _ {\mathfrak M } $ on $ M $ is defined by the system of semi-norms $ \{ \rho _ {A} \} $, $ A \in \mathfrak M $, where $ \rho _ {A} ( x) = \sup _ {m \in A } | \phi ( m, x) | $( cf. Semi-norm). This topology is locally convex if and only if $ \cup \mathfrak M $ is a total set, i.e. it generates (in $ L $ as a vector space) all of $ L $. The topology $ \tau _ {\mathfrak M } $ is called the topology of uniform convergence on the sets of $ \mathfrak M $.

The finest topology on $ M $ which can be defined in terms of the dual pairs $ ( L, M) $ is the topology of uniform convergence on weakly bounded subsets of $ L $. This is the topology $ \tau _ {\mathfrak M } $ where $ \mathfrak M $ is the collection of all weakly bounded subsets of $ L $, and it is called the strong topology on $ M $, for brevity.

References

[a1] G. Köthe, "Topological vector spaces" , 1 , Springer (1969)
How to Cite This Entry:
Strong topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Strong_topology&oldid=48878