Matrix Methods in Analysis

In this set of lecture notes, we present a culmination of results on infinite matrices which were evolved by the members of the Katowice Br-anch of the Mathematics Institute of the Polish Academy of Sciences. In the early history of functional analysis Hsliding humpH methods were used extensively to establish some of the early abstract results in functional analysis. For example, the first proofs of versions of the Uniform Boundedness Principle by Hahn and Banach and Hildebrand utilized sliding hump methods ([18], [39], [42], [35]).
Since Banach and Steinhaus gave a proof of the Uniform Boundedness Principle based on the Baire Category Theorem, category methods have proven to be very popular in treating various topics in functional analysis [19]. In recent times, there has been a return to Hsliding humpH methods in treating various topics in functional analysis and measure theory. For example, in [341 Diestel and Uhl use a lemma of Rosenthal ([64]) as an abstract sliding hump method to treat a variety of topics in vector measure theory.
In a somewhat similar fashion, the Antosik-Mikusinski Diagonal Theorem ([53], [21, [3], (91) can be considered to be an abstract sliding hump method and has been employed to treat a wide variety of topics in functional analysis and measure theory ([4], [5], [6], [91, [121, [531, [541, [561, (571). The Antosik-Mikusinski Diagonal Theorem is a result concerning infinite matrices and has proven to be quite effective in treating various topics that were previously treated by Baire category methods (see in particular the texts [12], [56]). These notes present a result concerning infinite matrices which is of an even simpler and more elementary character than the Diagonal Theorem, and which can still be used to treat a wide variety of topics in functional analysis and measure theory ([16]).