Hyperopic Strict Topologies on l?

Jaime Orrillo | Rudy Jose Rosas Bazan

**Keywords:**Hyperopic topology | chrages | locally convex topology

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.