Difference between revisions of "Weak topology"
From Encyclopedia of Mathematics
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| − | The [[ | + | {{TEX|done}} |
| + | |||
| + | 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==== | ||
| − | <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)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H.H. Schaefer, "Topological vector spaces" , Springer (1971)</TD></TR></table> | + | <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> | ||
====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 $F$ is a [[total set]], that is, separates the points of $X$. |
| − | See also [[ | + | See also [[Strong topology]]. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Jarchow, "Locally convex spaces" , Teubner (1981) (Translated from German)</TD></TR></table> | + | <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> | ||
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=17244
Weak topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weak_topology&oldid=17244
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article