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The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083690/s0836901.png" /> dual to the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083690/s0836902.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083690/s0836903.png" /> is the strong dual to a Hausdorff [[Locally convex space|locally convex space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083690/s0836904.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083690/s0836905.png" /> is equipped with the strong topology. Each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083690/s0836906.png" /> generates an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083690/s0836907.png" /> in accordance with the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083690/s0836908.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083690/s0836909.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083690/s08369010.png" />, the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083690/s08369011.png" /> is semi-reflexive. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083690/s08369012.png" /> is a [[Barrelled space|barrelled space]], the linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083690/s08369013.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083690/s08369014.png" /> is an isomorphic imbedding of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083690/s08369015.png" /> into the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083690/s08369016.png" />. The imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083690/s08369017.png" /> is called canonical. For normed spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083690/s08369018.png" /> is an isometric imbedding.
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The space  $  X  ^ {\prime\prime} $
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dual to the space  $  X  ^  \prime  $,
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where  $  X  ^  \prime  $
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is the strong dual to a Hausdorff [[Locally convex space|locally convex space]]  $  X $,
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i.e.  $  X  ^  \prime  $
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is equipped with the strong topology. Each element  $  x \in X $
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generates an element  $  F \in X  ^ {\prime\prime} $
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in accordance with the formula  $  F( f  ) = f( x) $(
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$  f \in X  ^  \prime  $).
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If  $  X  ^ {\prime\prime} = X $,
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the space  $  X $
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is semi-reflexive. If  $  X $
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is a [[Barrelled space|barrelled space]], the linear mapping  $  \pi :  X \rightarrow X  ^ {\prime\prime} $
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defined by  $  \pi ( x)= F $
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is an isomorphic imbedding of the space  $  X $
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into the space  $  X  ^ {\prime\prime} $.
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The imbedding  $  \pi $
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is called canonical. For normed spaces  $  \pi $
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is an isometric imbedding.
  
 
====Comments====
 
====Comments====
The second dual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083690/s08369019.png" /> is also called the bidual.
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The second dual $  X  ^ {\prime\prime} = ( X  ^  \prime  )  ^  \prime  $
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is also called the bidual.
  
For (semi-) reflexivity see also [[Reflexive space|Reflexive space]]. For the (first) dual space see [[Adjoint space|Adjoint space]]. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083690/s08369020.png" /> is reflexive if the canonical imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083690/s08369021.png" /> is surjective and also the two topologies coincide, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083690/s08369022.png" /> is given the [[Strong topology|strong topology]] defined by the dual pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083690/s08369023.png" />. For Banach spaces semi-reflexivity is the same as reflexivity.
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For (semi-) reflexivity see also [[Reflexive space|Reflexive space]]. For the (first) dual space see [[Adjoint space|Adjoint space]]. The space $  X $
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is reflexive if the canonical imbedding $  X \rightarrow X  ^ {\prime\prime} $
 +
is surjective and also the two topologies coincide, where $  X  ^ {\prime\prime} $
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is given the [[Strong topology|strong topology]] defined by the dual pair $  ( X  ^  \prime  , X  ^ {\prime\prime} ) $.  
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For Banach spaces semi-reflexivity is the same as reflexivity.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. van Dulst,  "Reflexive and superreflexive spaces" , ''MC Tracts'' , '''102''' , Math. Centre  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Köthe,  "Topological vector spaces" , '''1''' , Springer  (1969)  pp. §23.5</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. van Dulst,  "Reflexive and superreflexive spaces" , ''MC Tracts'' , '''102''' , Math. Centre  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Köthe,  "Topological vector spaces" , '''1''' , Springer  (1969)  pp. §23.5</TD></TR></table>

Latest revision as of 08:12, 6 June 2020


The space $ X ^ {\prime\prime} $ dual to the space $ X ^ \prime $, where $ X ^ \prime $ is the strong dual to a Hausdorff locally convex space $ X $, i.e. $ X ^ \prime $ is equipped with the strong topology. Each element $ x \in X $ generates an element $ F \in X ^ {\prime\prime} $ in accordance with the formula $ F( f ) = f( x) $( $ f \in X ^ \prime $). If $ X ^ {\prime\prime} = X $, the space $ X $ is semi-reflexive. If $ X $ is a barrelled space, the linear mapping $ \pi : X \rightarrow X ^ {\prime\prime} $ defined by $ \pi ( x)= F $ is an isomorphic imbedding of the space $ X $ into the space $ X ^ {\prime\prime} $. The imbedding $ \pi $ is called canonical. For normed spaces $ \pi $ is an isometric imbedding.

Comments

The second dual $ X ^ {\prime\prime} = ( X ^ \prime ) ^ \prime $ is also called the bidual.

For (semi-) reflexivity see also Reflexive space. For the (first) dual space see Adjoint space. The space $ X $ is reflexive if the canonical imbedding $ X \rightarrow X ^ {\prime\prime} $ is surjective and also the two topologies coincide, where $ X ^ {\prime\prime} $ is given the strong topology defined by the dual pair $ ( X ^ \prime , X ^ {\prime\prime} ) $. For Banach spaces semi-reflexivity is the same as reflexivity.

References

[a1] D. van Dulst, "Reflexive and superreflexive spaces" , MC Tracts , 102 , Math. Centre (1978)
[a2] G. Köthe, "Topological vector spaces" , 1 , Springer (1969) pp. §23.5
How to Cite This Entry:
Second dual space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Second_dual_space&oldid=48640
This article was adapted from an original article by M.I. Kadets (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article