Fuzzifying topologies on the space of linear operators

We introduce the notion of fuzzifying topologies on the space of linear operators. The concepts of fuzzifying topologies of pointwise convergence, bounded convergence, and complete bounded convergence are presented. We prove a sufficient and necessary condition for which the fuzzifying topologies are compatible with the structure on the space of linear operators. The Hausdorff separation of a fuzzifying topology on the space of linear operators is discussed. Specially, the net convergence, fuzzy boundedness, and fuzzy continuity of evaluation mappings in the fuzzifying topological linear space of pointwise convergence are investigated.