Reviewed techniques in automatic continuity of linear functionals

Document Type : Original Article


Department of Mathematics, Alagappa University, Karaikudi-630 004, India



Some techniques which were already used to derive automatic continuity results are chosen, they are modified, and extended results as well as generalized results are obtained. A technique of using the open mapping theorem and a technique of using the Hahn Banach extension theorem are explained. Results in connection with measurable cardinals are also obtained. Results for multiplicative linear functionals, positive linear functionals and uniqueness of topology are obtained. For example, sequential continuity of real multiplicative linear functionals on sequentially complete LMC algebras is obtained, when Michael's open problem is concerned only with boundedness of multiplicative linear functionals. The continuity of positive linear functionals on F-algebras with identity elements and involution is derived, when these functionals are continuous on the set of all involution-symmetric elements. Possibilities of extending the concept of positive linear functionals are considered to derive results for the continuity of such functionals on topological groups and topological vector spaces with additional structures. The technique for the Carpenter's uniqueness theorem is modified to derive boundedness of some homomorphisms. The entire article is oriented towards Michael's problem.