View source for Weak topology

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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====
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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>
<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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====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====
<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>
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