Weak Topologies of Normed Linear Spaces

Abstract

The concepts of weak convergence introduced by Banach [9, pp. 122, 133]1 are of exceeding interest, but in certain respects are rather restricted. Many of the results concerned with these concepts are in general valid only for separable spaces. It is natural, then, to turn to the general theory of limits of Moore and Smith [2] for a generalization. A unified theory of weak convergence is presented in Section 1, together with some of its consequences for weak convergence of functionals. Of these the principal result is Theorem 1:3, which states that the unit sphere of the adjoint of a Banach space is bicompact in the weak topology of functionals. This is a generalization of a result given by Banach [9, p. 123] which is valid only for separable spaces. Characterizations of spaces equivalent or isomorphic to adjoint spaces are given in Section 2, and are applied to a class of spaces considered by Dunford and Morse [13]. A result on universal spaces is given in Section 3. The remainder of the paper is devoted to the development of theories of integration and differentiation of abstract functions. Two theories of integration are outlined in Section 4, both of which can be obtained from that of Birkhoff [11] by replacing unconditional convergence of series by its analogues for weak convergence of functionals and elements. The second of these integrals can be specialized from that of Gelfand [15] by the addition of a uniformity condition. In Section 5 a theory of differentiation of functions with values in an adjoint space is developed. Instead of defining the as the limit of the difference quotients of the function, the class of all ultimate limit points of the difference quotients in the sense of weak convergence of functionals is considered. For a function f of bounded absolute variation in the sense of Definition 5:1, it is shown that these sets are non-null almost everywhere, so that they define a multiple-valued function, the derivative of f. By using the first of the integrals defined in Section 4, it is shown that the fundamental theorem of the calculus holds for of absolute continuous functions. Gelfand's result on weak derivatives [15], and the results of Clarkson [12], Dunford and Morse [13], Gelfand [15], and Pettis [18] on strong derivatives are derived from this result. It is of interest to note that in the derivation of Clarkson's theorem, it is shown that uniformly convex spaces are regular.