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Difference between revisions of "Weak topology"

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The [[Locally convex topology|locally convex topology]] on a vector space $X$ generated by the family of semi-norms (cf. [[Semi-norm|Semi-norm]]) $p(x)=|f(x)|$, where $f$ ranges over some subset $F$ of the (algebraic) [[Adjoint space|adjoint space]] $X^*$.
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The [[locally convex topology]] on a vector space $X$ generated by the family of [[semi-norm]]s $p(x)=|f(x)|$, where $f$ ranges over some subset $F$ of the (algebraic) [[adjoint space]] $X^*$.
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.A. Lyusternik,  V.I. Sobolev,  "A short course of functional analysis" , Moscow  (1982)  (In Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H.H. Schaefer,  "Topological vector spaces" , Springer  (1971)  {{MR|0342978}} {{MR|0276721}} {{ZBL|0217.16002}} {{ZBL|0212.14001}} </TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  L.A. Lyusternik,  V.I. Sobolev,  "A short course of functional analysis" , Moscow  (1982)  (In Russian)  {{MR|}} {{ZBL|}} </TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  H.H. Schaefer,  "Topological vector spaces" , Springer  (1971)  {{MR|0342978}} {{MR|0276721}} {{ZBL|0217.16002}} {{ZBL|0212.14001}} </TD></TR>
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</table>
  
  
  
 
====Comments====
 
====Comments====
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The weak topology as introduced above is often denoted by $\sigma(X,F)$. It is a Hausdorff topology if and only if $F$ separates the points of $X$.
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The weak topology as introduced above is often denoted by $\sigma(X,F)$. It is a Hausdorff topology if and only if $F$ is a [[total set]], that is, separates the points of $X$.
  
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See also [[Strong topology|Strong topology]].
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See also [[Strong topology]].
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Jarchow,  "Locally convex spaces" , Teubner  (1981)  (Translated from German)  {{MR|0632257}} {{ZBL|0466.46001}} </TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Jarchow,  "Locally convex spaces" , Teubner  (1981)  (Translated from German)  {{MR|0632257}} {{ZBL|0466.46001}} </TD></TR>
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</table>

Latest revision as of 22:17, 10 December 2016


The locally convex topology on a vector space $X$ generated by the family of semi-norms $p(x)=|f(x)|$, where $f$ ranges over some subset $F$ of the (algebraic) adjoint space $X^*$.

References

[1] L.A. Lyusternik, V.I. Sobolev, "A short course of functional analysis" , Moscow (1982) (In Russian)
[2] H.H. Schaefer, "Topological vector spaces" , Springer (1971) MR0342978 MR0276721 Zbl 0217.16002 Zbl 0212.14001


Comments

The weak topology as introduced above is often denoted by $\sigma(X,F)$. It is a Hausdorff topology if and only if $F$ is a total set, that is, separates the points of $X$.

See also Strong topology.

References

[a1] H. Jarchow, "Locally convex spaces" , Teubner (1981) (Translated from German) MR0632257 Zbl 0466.46001
How to Cite This Entry:
Weak topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weak_topology&oldid=39956
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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