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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