# Weak topology

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

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 $X$.