A Simple Theory of Differential Calculus in Locally

A theory of differential calculus for nonlinear maps between general locally convex spaces is developed. All convergence notions are topological, and only familiarity with basic results from point set topology, differential calculus in Banach spaces, and locally convex space theory is assumed. The chain rule for continuous kth order differentiability, smoothness of inverse functions, and function space continuity properties of higher order derivatives are examined. It is shown that this theory extends the classical Fréchet theory of differential calculus for maps between Banach spaces. Introduction. An open question for several decades has been whether it is possible to develop a "nice" theory of Ck differentiability for maps between general locally convex spaces which extends the standard theory of Fréchet differential calculus for maps between Banach spaces. Such a theory should have several properties: (1) The class of Ck maps should be closed under composition. (2) Continuous linear and multilinear maps between locally convex spaces should beC00. (3) The theory should establish a framework for the study of inverse functions, local flows for Ck vector fields, and C00 partitions of unity. (4) Proofs and concepts should be simple, and close to those from Banach space Fréchet calculus. We take an indirect approach to the solution of this problem. In [3], the author developed an alternative approach to Banach space differential calculus, in which derivatives are continuous with respect to the strong operator topology on spaces of linear and multilinear maps rather than the uniform operator topology. However, this theory (called Sfk calculus) was shown to be closely related to Ck calculus, and the notions of £fk and Ck differentiability were shown to coincide in the case k — oo. In this article, the notion of y * differentiability is extended to the general locally convex setting. The chain rule is established for y* maps, continuous linear and multilinear maps are shown to be y00, and y1 inverses of y * maps are shown to be y* differentiable. After the development of basic y* calculus, Ck differentiability is defined in terms of y* differentiability, and the corresponding results are established for Ck calculus. It is shown that this Ck calculus extends the notion of Banach space Received by the editors July 27, 1984. 1980 Mathematics Subject Classification. Primary 58C20. 46G05; Secondary 46A05, 47H99. ©1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page 485 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use