We explicitly define  a family of seminorms    on   the space  of all  bounded real sequence  $l^{\infty}.$  This family   gives rise to  a  Hausdorff locally convex topology  which is not equivalent to  the usual ones:    the weak topology $\sigma( l^{\infty}, l^1),$  the norm topology $\tau_{\infty},$ the Mackey topology $m(l^{\infty}, l^1)$ and the strict  topology $\beta.$   We show that this new topology, denoted by $\beta_h,$  is weaker than the norm topology, $\tau_{\infty}.$ Finally, we show that the   dual of  $l^{\infty}$  with respect to  $\beta_h,$ called hyperopic strict dual,  is not  $l^1$  anymore but rather,  is identified with  the set of all purely    finitely additive measures.