Rings of continuous linear transformations on a topological vector space

A structural analysis of the ring CL(X) of all linear continuous transformations on a topological vector space X is presented here. It explores the algebraic cum topological properties of CL(X) in relation with the structural features of X. Specifically, we discussed the relations between left ideals, zero-sets and z-filters in a topological vector space X and proved that if X is a finite dimensional topological vector space, the left ideal generated by the Zero set of a left ideal I in CL(X) is I itself.