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Difference between revisions of "Adjoint space"

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''of a topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010890/a0108901.png" />''
 
  
The vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010890/a0108902.png" /> consisting of continuous linear functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010890/a0108903.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010890/a0108904.png" /> is a locally convex space, then the functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010890/a0108905.png" /> separate the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010890/a0108906.png" /> (the Hahn–Banach theorem). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010890/a0108907.png" /> is a normed space, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010890/a0108908.png" /> is a Banach space with respect to the norm
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{{TEX|done}}
  
<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/a/a010/a010890/a0108909.png" /></td> </tr></table>
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''of a topological vector space $E$''
  
There are two (usually different) natural topologies on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010890/a01089010.png" /> which are often used: the strong topology determined by this norm and the weak-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010890/a01089011.png" />-topology.
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The vector space $E^{*}$ consisting of continuous linear functions on $E$. If $E$ is a locally convex space, then the functionals $f\in E^{*}$ separate the points of $E$ (the [[Hahn–Banach_theorem | Hahn–Banach theorem]]). If $E$ is a normed space, then $E^{*}$ is a Banach space with respect to the norm
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\begin{equation*}
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\|f\| = \sup\limits_{x\ne0}\frac{|f(x)|}{\|x\|}.
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\end{equation*}
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There are two (usually different) natural topologies on $E^{*}$ which are often used: the strong topology determined by this norm and the weak-$*$-topology.
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
Instead of the term adjoint space one more often uses the term dual space. The weak-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010890/a01089013.png" />-topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010890/a01089014.png" /> is the weakest topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010890/a01089015.png" /> for which all the evaluation mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010890/a01089016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010890/a01089017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010890/a01089018.png" />, are continuous.
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Instead of the term adjoint space one more often uses the term dual space. The weak-$*$-topology on $E^{*}$ is the weakest topology on $E^{*}$ for which all the evaluation mappings $f\mapsto f(x)$, $f\in E^{*}$, $x\in E$, are continuous.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.H. Schaefer,  "Topological vector spaces" , Macmillan  (1966)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.H. Schaefer,  "Topological vector spaces" , Macmillan  (1966)</TD></TR></table>

Latest revision as of 18:01, 30 November 2012


of a topological vector space $E$

The vector space $E^{*}$ consisting of continuous linear functions on $E$. If $E$ is a locally convex space, then the functionals $f\in E^{*}$ separate the points of $E$ (the Hahn–Banach theorem). If $E$ is a normed space, then $E^{*}$ is a Banach space with respect to the norm \begin{equation*} \|f\| = \sup\limits_{x\ne0}\frac{|f(x)|}{\|x\|}. \end{equation*} There are two (usually different) natural topologies on $E^{*}$ which are often used: the strong topology determined by this norm and the weak-$*$-topology.

References

[1] D.A. Raikov, "Vector spaces" , Noordhoff (1965) (Translated from Russian)

Comments

Instead of the term adjoint space one more often uses the term dual space. The weak-$*$-topology on $E^{*}$ is the weakest topology on $E^{*}$ for which all the evaluation mappings $f\mapsto f(x)$, $f\in E^{*}$, $x\in E$, are continuous.

References

[a1] H.H. Schaefer, "Topological vector spaces" , Macmillan (1966)
How to Cite This Entry:
Adjoint space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adjoint_space&oldid=17289