The starting point for this investigation was an attempt to generalize the well known theorem of Silverman and Toeplitz [25](1) on regularity of a sequence-to-sequence transformation. This theorem may be stated as follows: If a transformation of sequences { t, } of real numbers to sequences { Sm } is defined from a matrix { amn }, m, n = 1, 2, * , by the equations Sm =ZEn amntn, the transformation is regular-that is, is defined everywhere and takes every convergent sequence { t, } into another convergent sequence with the same limit-if and only if the matrix { amn } satisfies the conditions (a) limm dn 'tamn = 1, (b) limm amn = 0 for each n, and (c) there is a K such that En I amn, < K for every m. In the special case under consideration, the fact that regularity implies condition (c) (the non-trivial part of the proof) can be derived from a theorem of Banach [3, p. 80, Theorem 5 ]: If A and B are Banach spaces, and if U, n = 1, 2, * * , are. linear operators on A to B, such that lim supn || Un(a)ll < o? for each a in A, then lim supn 11 U711 < oo If the sequence of integers is replaced by a directed set X, it is known that A, B, X, and U. can be chosen for which the similar statement relating lim supx || Ux(a)ll and lim supx || Ux,l is false; sections 1-3 of this paper consider these cases in an attempt to solve the problem of boundedness: Characterize those Banach spaces A and B, and directed sets X such that choosing the linear operators U, on A to B so that lim supx || U,(a)ll < oo for each a in A implies that lim supx || Ux|| < oo. Section 1 is a review of pertinent facts about directed sets and convergence (mostly due to Moore and Smith [19], G. Birkhoff [5], and Tukey [27]). Section 2 studies the relations among three topologies in the space of operators on A to B. In ?3 the problem of boundedness is studied but not completely solved. The second part of the paper is concerned with certain special operators on some function spaces. In ?4 the space is that of the totally measurable functions on a set Y to a Banach space B; a class of operators on this space is defined in terms of additive, real-valued set-functions and the relations among various topologies in this set of operators is given; this is used in ?6 to give a general form to a theorem of Vulich [28]. In ?5 the functions studied are the measurable functions on Y to B; the operations on this space are defined in terms of completely additive, limited, set-functions whose values are transfor-