The dual of linear topological space

In this work, locally convex linear topological space  with dual space are presented for the investigation of the existence and properties of various unique topologies on the dual space . We established the existence of three distinct topologies on generated by polar of subsets of Uses were made of the definition of base and subbase of topology and properties of locally convex Hausdorff topology determined by a separating family  of seminorms having base at zero. We have also established that the three distinct topologies on  are convex, absorbent and Hausdorff, using the concept of seminormed linear topology. Not alone, we established that the point open topology is less than the compact open topology while the compact open topology is less than the strong topology. And that the compact open topology coincides with the strong topology whenever is of finite dimensional. Use was made of the definition and concepts of coarser topology.