Linear Transformations

The purpose of this paper is to give a characterization of linear and completely continuous transformations both on the common Banach spaces to an arbitrary Banach space and vice versa. There is an abundant literature on this subject. Among the earliest papers, the now famous paper of Radon [241 should be mentioned. Here linear transformations on LP to Lq (1 <p, q< oc) are characterized in a manner suggestive of the methods used in the present paper. The works of Gelfand [12], Dunford [6], Kantorovitch and Vulich [17], and Dunford and Pettis [9] contain much material on this subject supplementary to that treated here. In the interest of completeness we have restated a few of the results obtained by Gelfand [12 ], and Gowurin [13 ]. The principal tools used in our characterizations are certain abstractly valued function spaces. One such space is the class of all additive set functions x(r) on all Lebesgue measurable subsets r of (0, 1) to a Banach space X where for all linear functionals x on X and for all subdivisions 7r=` (Irl, T2, , *r ) of (0, 1) into disjoint sets,