Implicitly defined mappings in locally convex spaces

Terrence S. McDermott
1971 Transactions of the American Mathematical Society  
Results on existence, uniqueness, continuity and differentiability of implicit functions in locally convex, linear topological spaces are obtained, and certain of these results are applied to obtain results on the existence and continuous dependence on parameters of global solutions for a nonlinear Volterra integral equation. Introduction. So far, no single theory of differentiation or local linear approximation of nonlinear mappings between locally convex, linear topological spaces seems to
more » ... spaces seems to have clearly established a place of preeminence. In particular, none has produced an implicit function theorem of anywhere near the comprehensive character of the results available in the context of Banach spaces (see, for example, [3] ). Results in more general settings can be found in [1], [4], [5], and [6]. In this article, we discuss the problem of implicit functions in locally convex spaces in light of the ideas and results developed in [11] . In §1 we give conditions for the existence of an implicit function, then obtain results dealing with its uniqueness and continuity. In §2 certain results from the first are applied to discuss global, continuous solutions for a nonlinear, Volterra integral equation. Finally, in §3 we give conditions under which a differentiable implicit function will exist, differentiability being understood in the sense of Sebastiào è Silva [14] . For the convenience of the reader, §0 has been included containing the essential definitions and results from [11] that are used in the present work. 0. Preliminaries. In this section, we shall state the basic definitions and results from [11] that will be needed in the present work. Throughout the section, let E and F denote real, locally, convex, linear topological spaces. If A is an absolutely convex set in E, we denote by EA the linear subspace of E generated by A. The functional ||-||A defined on EA by ||jc||A = inf {A>0 : xeXA} is a norm on EA in case £ is Hausdorff and A is bounded. Further, ||x|[ayl = (l/a)||x||/1 for any a > 0. We assume henceforth that all spaces are Hausdorff.
doi:10.1090/s0002-9947-1971-0283536-2 fatcat:d6lm36wjnjdttihvuyjrcfehyu