On bounded linear functional operations

Abstract

The set of all bounded linear functional operations on a given vector spacet plays an important role in the consideration of linear functional operations. For the sake of greater definiteness it is desirable to know the form of such operations and a space determined thereby. This problem has been solved for a number of spaces, for instance, all continuous functions on a bounded closed interval, all Lebesgue pth power (p > 1) integrable functions, all sequences whose pth powers (p > 1) form absolutely convergent series, all sequences having a limit, and so on.T All of these spaces have the property of separability. For non-separable spaces, there is a recent determination of the operation for the space of all bounded functions on a finite interval, having at most discontinuities of the first kind, by H. S. Kaltenborn. ? In this paper we give a determination of the linear operation for the space of (a) all bounded sequences, (b) all bounded measurable functions, (c) all bounded functions on the infinite interval having at most discontinuities of the first kind, (d) all bounded continuous functions on the infinite interval, (e) all almost bounded functions. With the least upper bound as norm, all of these spaces are not separable. 1. Notations. The integral. We shall denote by (a) $ a set of elements p. (b) t a real-valued function on $. (c) X a set of functions t. (d) C a set or class of subsets E of $, containing the null set and the set $.