Hamilton R S, The Inverse Function Theorem Of Nash And Moser, Bulletin Of Ams, 1982.pdf

BULLETIN (New Series) OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 7, Number 1, July 1982
THE INVERSE FUNCTION THEOREM
OF NASH AND MOSER
BY RICHARD S. HAMILTON
Contents
Introduction
Part I. Calculus on Fréchet spaces
. Fréchet spaces
. Definition of a Fréchet space
. Properties of Fréchet spaces
. Families of linear maps
. The Riemann integral
. The definite integral
. Parametrized curves
. The directional derivative
. Definition of the directional derivative
. Properties of the derivative
. The chain rule
. Partial derivatives
. Second derivatives
. Higher derivatives
. Fréchet manifolds
. Manifolds
. Submanifolds
. Vector bundles
. Maps of manifolds
. Connections
. Lie groups
. The inverse function theorem
. Estimates
. The inverse function theorem for Banach spaces
. Inverses of linear maps
. Examples in Banach spaces
. Counterexamples in Fréchet spaces
. Differential equations in Fréchet spaces
Part II. The Nash-Moser Category
II. 1. Tame Fréchet spaces
. Graded Fréchet spaces
. Tame linear maps
. Tame Fréchet spaces
Received by the editors November 15, 1981.
1980 Mathematics Subject Classification. Primary 58C15; Secondary 58C20, 58D05, 58G30.
© 1982 American Mathematical Society
0273-0979/82/0000-0335/$
65
66 R. S. HAMILTON
. Tame maps
. Definition of a tame map
. Partial differential operators
. Tame Fréchet manifolds
. Inverses of families of linear maps
. Smooth tame inverses
. Ordinary differential equations
. Elliptic equations
. Symmetric systems
Part III. The Nash-Moser theorem
. The proof
. Statement of the theorem
. Normalizations
. Injectivity
. Smoothing operators
. Surjectivity
. A priori estimates
. The inverse map