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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097250/w0972501.png" /> generated by the family of semi-norms (cf. [[Semi-norm|Semi-norm]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097250/w0972502.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097250/w0972503.png" /> ranges over some subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097250/w0972504.png" /> of the (algebraic) [[Adjoint space|adjoint space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097250/w0972505.png" />.
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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]] $X^*$.
  
 
====References====
 
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====Comments====
 
====Comments====
The weak topology as introduced above is often denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097250/w0972506.png" />. It is a Hausdorff topology if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097250/w0972507.png" /> separates the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097250/w0972508.png" />.
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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 $X$.
  
 
See also [[Strong topology|Strong topology]].
 
See also [[Strong topology|Strong topology]].

Revision as of 15:38, 13 July 2014

The locally convex topology on a vector space $X$ generated by the family of semi-norms (cf. Semi-norm) $p(x)=|f(x)|$, 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 $F$ 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=28283
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article