View source for Weak topology

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" />.

====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)  {{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====
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" />.

See also [[Strong topology|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></table>