somewhat more complicated than the topological approach presented
here in Theorem 1.25. The uu-construction of Theorem 1.25, which
associates a globally-defined weaker topology to a Banach manifold
of maps under much more general conditions than those original
methods, was something I noticed a week after the initial discovery
by the above-mentioned researchers. In fact, it was an attempt to
find an intuitive justification for the existence of the topology
discovered by Uhlenbeck and Dowling which led me to the
^-construction and, later, to the material in Chapter 2.
I would like to acknowledge the contributions of several in-
dividuals who provided valuable information at several stages in the
development of this material. R. R. Phelps and J. H. C. Whitfield
provided me with capsule summaries of the basic facts about
differentiable norms on Banach spaces which I needed to extend the
critical point theory of Chapter 8 from Hilbert manifolds to more
general Banach manifolds. J. p. Penot developed the proof of
Lemma 5.25 presented here, which was necessary for the extension of
the inverse function theorem of Chapter 5 from Bw* spaces to general
bw* spaces (my original proof of the inverse function theorem for Bw*
spaces only required a proof of Lemma 5.25 for metric spaces).
J. R. Dorroh helped me simplify and extend the proof of Lemma 5.lU,
which I had originally proved for bw* spaces, so that it now applies
to all locally convex spaces. And, most significantly of all,
R. S. Palais provided me with suggestions, insights about global
analysis, and general encouragement during the period 1969-1972, when
I was a graduate student and much of the development of this theory
took place.
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