In locally convex space theory,there is a nice result due to N.J.Kalton as follows(see [1] and [2]):Let (E,F) be a dual pair of vector spaces and suppose that τis an (E,F)-polartopology on E such that (E,τ) is separable.Then if ∑x_j is σ(E,F)-subseriesconvergent,then it is τ-subseries convergent.
Recently, the locally convex space theory has obtained a series of proper developments and improvements by the agency of the Basic Matrix Theorem (BMT) duc to J. Mikusinski and P. Antosik. In this note, we would like to present another basic theorem named Uniform Convergence Principle (UCP). We shall show that UCP has the same effects as BMT, though UCP is easier than BMT in their proofs. UCP. Let G be an abelian topological group and Ωa sequentially compact space.
