  Let $A$ be a commutative and unital $\mathbb{R}$-algebra, and $M$ be an
Archimedean quadratic module of $A$. We define a submultiplicative seminorm
$\|\cdot\|_M$ on $A$, associated with $M$. We show that the closure of $M$ with
respect to $\|\cdot\|_M$-topology is equal to the closure of $M$ with respect
to the finest locally convex topology on $A$. We also compute the closure of
any cone in $\|\cdot\|_M$-topology. Then we omit the Archimedean condition and
show that there still exists a lmc topology associated to $M$, pursuing the
same properties.
