On Some Locally Convex Topologies of a Vector Space

On Some Locally Convex Topologies of a Vector Space

Suppose that $(X,\tau)$ is already a locally convex TVS. Let us denote by
$X'$, the space of all $\tau$-continuous linear functionals on $X$, the
topological dual of $X$. For each $f\in X'$, define $p_f(x)=|f(x)|, x\in
X$. Then $$\Gamma=\{p_f: f\in X'\}$$ is a separating family of seminorms
on $X$. By a theorem of Rudin, there exists a locally convex topology on
$X$, commonly denoted by $\sigma:=\sigma(X,X')$, with the property that
every member of $\Gamma$ is $\sigma$-continuous. Specifically, $\sigma$
consists of arbitrary unions of the translates of finite intersections of
the sets $\{x\in X: |f(x)|<\epsilon\}$, $f\in X'$ and $\epsilon>0$.
Question. Is $\tau=\sigma$? What is only clear to me (if Im correct) is
that $\sigma\subset\tau$. I need some help on this.