The Geometry of Banach Spaces. Smoothnesso'2)

Introduction. This paper contains the first unified treatment of the dual theory of differentiability of the norm functional in a real normed linear space. With this, the work of Smulian [2; 3] is extended and it is shown how uniform convexity is to be modified so as to obtain geometric properties dual to the various types of differentiability of the norm thus answering a question implicit in the work of Lovaglia [1] and Anderson [1]. The resulting dual theory of differentiability is then used to obtain a connection between the differentiability of certain infinite dimensional manifolds imbedded in an infinite dimensional real normed linear space and the continuity properties of the Gaussian spherical image map generalized to such infinite dimensional spaces. In this way a problem proposed by Klee [2, p. 35] is solved. The principal tools employed are (a) Mazur's characterization [1] of a supporting hyperplane of the unit ball as an inverse image of the derivative of the norm functional; (b) a modification of an integral calculus technique given in Krasnosel'skn and Rutickn[l,p. 187]; and (c) James' criterion for the reflexivity of a Banach space, viz., that every continuous linear functional attains its supremum on the unit ball. James [1] has given a proof of this criterion for separable Banach spaces and in James [2] has removed the condition that the Banach space be separable. The paper is divided into five sections. §1 contains the localization and directionalization of uniform convexity of Clarkson [1] and of full fc-convexity of Fan and Glicksberg [1]. §2, while containing interesting facts in its own right, is motivational in nature for §3. §3 contains in the dual theory of differentiability of the norm one of the two main conclusions of the paper. The other main conclusion contained in §4 is the analysis of the differentiability of norm functionals using the generalized Gaussian spherical image map. The last section, §5, examines and compares the present results in the context of the geometry of Banach spaces as created and perfected by other workers in the field.