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 ...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.展开更多
The authors obtain new characterizations of unconditional Cauchy series in terms of separation properties of subfamilies of p(N), and a generalization of the Orlicz-Pettis Theorem is also obtained. New results on the ...The authors obtain new characterizations of unconditional Cauchy series in terms of separation properties of subfamilies of p(N), and a generalization of the Orlicz-Pettis Theorem is also obtained. New results on the uniform convergence on matrices and a new version of the Hahn-Schur summation theorem are proved. For matrices whose rows define unconditional Cauchy series, a better sufficient condition for the basic Matrix Theorem of Antosik and Swartz, new necessary conditions and a new proof of that theorem are given.展开更多
文摘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.
文摘The authors obtain new characterizations of unconditional Cauchy series in terms of separation properties of subfamilies of p(N), and a generalization of the Orlicz-Pettis Theorem is also obtained. New results on the uniform convergence on matrices and a new version of the Hahn-Schur summation theorem are proved. For matrices whose rows define unconditional Cauchy series, a better sufficient condition for the basic Matrix Theorem of Antosik and Swartz, new necessary conditions and a new proof of that theorem are given.
