Elements of nonlinear functional analysis

Appropriate locally convex domains for differential calculus

“…A natural question to ask is whether there exists an easily-characterized class of D-spaces to which both of these classes belong. We answer this question in the affirmative in Theorem 1 below, the proof of which presents a much clearer picture of the nature of the key property of Z)-spaces than the corresponding result [2,Theorem 3.46].…”

“…Introduction. In [2], the first named author introduced a theory of differential calculus in locally convex spaces. This theory differs from previous approaches to the subject in that the theory was an attempt to isolate a class of locally convex spaces to which the usual techniques of Banach space differential calculus could be extended, rather than an attempt to develop a theory of differential calculus for all locally convex spaces.…”

Book Review: The metric theory of Banach manifolds

“…The first approach has been pursued in some detail by the reviewer [6] in connection with applications to the calculus of variations; the second approach has been taken by many authors-indeed, by anyone who has ever investigated dif-ferential calculus in Fréchet spaces, since the study of C CC (M V M 2 ) reduces essentially to the study of the analytical structure of the transition functions on the model space. The most exciting work of the latter type has been done by R. Hamilton.…”

“…And while flows for these equations have been constructed by regarding the equations as unbounded vector fields on appropriate Banach function spaces (see, in particular, [10], [11], [18]), these techniques require the use of at least two distinct manifolds in the function space scale. A determination of the relation of these theorems to the inverse function theorems of [7], [8], or [6] In connection with the topic of flows for unbounded operators, it is appropriate to mention a result of J. Marsden [13], who has constructed a flow for the sum of a bounded and an unbounded operator from the flows for the individual operators by a Lie product technique, something like P. Chernoff's nonlinear generalization of the Trotter product formula. However, the theorems of Chernoff and Trotter are very different in that they show the convergence of a Lie product approximation scheme to a flow which is already assumed to exist.…”

“…For example, the theorems on flows for unbounded operators cited above, and the usual existence and uniqueness theorem for flows generated by locally Lipschitz vector fields on Banach spaces, both reduce to the same result for vector fields on finite-dimensional spaces, though in infinite dimensions the theorems for unbounded vector fields are much more general (modulo the fact that the theorems of [10] and [11] apply only to reflexive spaces). As another example, compare the inverse function theorem of [6] with the usual inverse function theorem for C k mappings between Banach spaces: again, the two theorems reduce to the same result in finite dimensions, though for infinite-dimensional spaces the second result is more general but with a weaker conclusion than the first.…”