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Abstract

We prove that the Quasi Differential of Bayoumi of maps between locally bounded F-spaces may not be Fréchet-Differential and vice versa. So a new concept has been discovered with rich applications (see [1–6]). Our F-spaces here are not necessarily locally convex

. Soc. 73 (1979), 351–355. [8] Roberts J.W., A nonlocally convex F -space with the Hahn-Banach approximation prop- erty, in: Banach spaces of analytic functions (Proc. Pelczynski Conf., Kent State Univ., Kent, Ohio, 1976), Lecture Notes in Math., Vol. 604, Springer, Berlin, 1977, pp. 76–81. Włodzimierz Fechner 2. Remark (On the essential union and intersection of families of measur- able sets) The notions that we discuss below turn out to be significant if one tries to extend the Radon–Nikodym Theorem for non-σ-finite measure spaces. For details, we refer to Section 1

operators. We understand that some of the readers may not be interested in the non-Banach case. However, some signifi- cant applications to the isomorphic structure concern the nonlocally convex F-spaces, (these are contained in Chapter 3, and do not play a significant role in other chapters). Our main reference in this direction is Rolewicz’s monograph [122]. 1.2 Terminology and notation Throughout the book, we consider Köthe function spaces (for definitions see below) on finite atomless measure spaces only. We concentrate on atomless spaces since, by definition, a

nonisomorphic Banach spaces, Is- rael J. Math. 48 (1984), pp. 139–147. [121] J. W. Roberts, Pathological compact convex sets in Lp(0, 1), 0 < p < 1, The Altgeld book, University of Illinois Functional Analysis Seminar, 1975-76. [122] , A compact convex set with no extreme points, Studia Math. 60 (1977), pp. 255–266. [123] , A nonlocally convex F -space with the Hahn-Banach approximation property, Ba- nach spaces of analytic functions (Proc. Pelczynski Conf., Kent State Univ., Kent, Ohio, 1976). Lecture Notes in Math. 604, Springer, Berlin, 1977, pp. 76–81. [124] R. Rochberg