Difference between revisions of "Weak topology"
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| − | The [[Locally convex topology|locally convex topology]] on a vector space | + | {{TEX|done}} |
| + | 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==== | ====References==== | ||
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====Comments==== | ====Comments==== | ||
| − | The weak topology as introduced above is often denoted by | + | 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=32431
Weak topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weak_topology&oldid=32431
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article