The Bounded Consistency Theorem

Introduction. The Bounded Consistency Theorem (BCT, Theorem 1.2 below) is a principal result of summability theory. Indeed, it holds a claim to a second position in the theory just after the Silverman-Toeplitz Theorem (Theorem 1.1 below). My purpose here is to make the BCT and a proof of it accessible to the largest possible readership. If you have taken a course in functional analysis, you should find the narrative easy to follow, and a nice application of what you have learned. In fact, I hope that this article will be accessible to many who have the prerequisites for reading the second part of [11], i.e., a good course in advanced calculus and some linear algebra. Such a reader can preview some results of functional analysis and see how they apply to summability theory. Of course the harder you must work to read this article, the more you can expect to benefit from it. The expert on summability theory need only browse through the proof of Lemma 3.2 in order to grasp the main thrust of our development. This proof, based on the Schur Property of 11, is original, and apparently the easiest functional analytic proof. The basis for our discussion will be the text of Wilansky [17]. Remarkable for its history as well as its content, the Bounded Consistency Theorem was first announced without proof by S. Mazur and W. Orlicz [7] in 1933. Their investigation on the application of functional analysis to summability theory was a portion of the intense activity by the "school of Lwow," a group of functional analysts flourishing around S. Banach prior to the outbreak of World War II. A special case of the BCT had already appeared on page 95 of Banach's famous monograph [1]. Mazur and Orlicz were not able to publish a complete report of their investigation until 1954 [8]. Meanwhile, the BCT had been independently rediscovered by the Russian mathematician A. L. Brudno [3] and published with a different (nonfunctional analytic) proof requiring seven pages of calculations. In his 1955 book [4] R. G. Cooke described the work of Brudno on pages 130-131 and asked whether a shorter proof of the BCT could be constructed. This challenge was met by P. Erdos and G. Piranian writing jointly [5] and by G. M. Petersen [10]. Both of these proofs are streamlined versions of Brudno's. Petersen attacked the problem directly and completed the calculations in one page. Erdos and Piranian developed a general technique based on the methods of Brudno and called it "the principle of aping sequences." They used this technique to study in depth the bounded convergence field of a matrix, and derived the BCT as a quick application. The BCT is stated on page 67 of Zeller's monograph on summability theory [18]. Other proofs of the BCT or versions thereof can be found in [2], [9], [13] and [19]. We shall now explain our notation and some elements of summability theory leading up to the statement of the BCT. The jth coordinate of a sequence s of real or complex numbers is written s(j); thus s = (s(j)). We prefer the functional to the subscript notation because we often need to discuss sequences of sequences. The sum and scalar multiple of sequences are defined coordinatewise. For any sequence s, s[ n; s[ >n] = s s[ < n]. The sequence with 1 in the nth place and 0's elsewhere is denoted by e,. The space of all sequences with only finitely many nonzero coordinates is denoted by 0. The spaces of sequences which are bounded, convergent, or convergent to zero are denoted, respectively, by m, c and c0. The i, j element of an infinite matrix is denoted by A(i,j). For any matrix A, we denote by CA