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As long as a branch of knowledge offers an abundance of problems,
it is full of vitality. David Hilbert Over the last 15 years I have
given lectures on a variety of problems in nonlinear functional
analysis and its applications. In doing this, I have recommended to
my students a number of excellent monographs devoted to specialized
topics, but there was no complete survey-type exposition of
nonlinear functional analysis making available a quick survey to
the wide range of readers including mathematicians, natural
scientists, and engineers who have only an elementary knowledge of
linear functional analysis. I have tried to close this gap with my
five-part lecture notes, the first three parts of which have been
published in the Teubner-Texte series by Teubner-Verlag, Leipzig,
1976, 1977, and 1978. The present English edition was translated
from a completely rewritten manuscript which is significantly
longer than the original version in the Teubner-Texte series. The
material is organized in the following way: Part I: Fixed Point
Theorems. Part II: Monotone Operators. Part III: Variational
Methods and Optimization. Parts IV jV: Applications to Mathematical
Physics. The exposition is guided by the following considerations:
(a) What are the supporting basic ideas and what intrinsic
interrelations exist between them? (/3) In what relation do the
basic ideas stand to the known propositions of classical analysis
and linear functional analysis? ( y) What typical applications are
there? Vll Preface viii Special emphasis is placed on motivation.
This book is the first volume of a two-volume work, which is an
improved version of a preprint [47] published in German. We seek to
deduce the funda mental concepts of quantum mechanics solely from a
description of macroscopic devices. The microscopic systems such as
electrons, atoms, etc. must be detected on the basis of the
macroscopic behavior of the devices. This detection resembles the
detection of the dinosaurs on the basis offossils. In this first
volume we develop a general description of macroscopic systems by
trajectories in state spaces. This general description is a basis
for the special de scription of devices consisting of two parts,
where the first part is acting on the second. The microsystems are
discovered as systems transmitting the action. Axioms which
describe general empirical structures of the interactions between
the two parts of each device, give rise to a derivation of the
Hilbert space structure of quantum mechanics. Possibly, these
axioms (and consequently the Hilbert space structure) may fail to
describe other realms than the structure of atoms and mole cules,
for instance the "elementary particles". This book supplements ref.
[2]. Both together not only give an extensive foundation of quantum
mechanics but also a solution in principle of the measuring
problem.
As long as a branch of knowledge offers an abundance of problems,
it is full of vitality. David Hilbert Over the last 15 years I have
given lectures on a variety of problems in nonlinear functional
analysis and its applications. In doing this, I have recommended to
my students a number of excellent monographs devoted to specialized
topics, but there was no complete survey-type exposition of
nonlinear functional analysis making available a quick survey to
the wide range of readers including mathematicians, natural
scientists, and engineers who have only an elementary knowledge of
linear functional analysis. I have tried to close this gap with my
five-part lecture notes, the first three parts of which have been
published in the Teubner-Texte series by Teubner-Verlag, Leipzig,
1976, 1977, and 1978. The present English edition was translated
from a completely rewritten manuscript which is significantly
longer than the original version in the Teubner-Texte series. The
material is organized in the following way: Part I: Fixed Point
Theorems. Part II: Monotone Operators. Part III: Variational
Methods and Optimization. Parts IV jV: Applications to Mathematical
Physics. The exposition is guided by the following considerations:
(a) What are the supporting basic ideas and what intrinsic
interrelations exist between them? (/3) In what relation do the
basic ideas stand to the known propositions of classical analysis
and linear functional analysis? ( y) What typical applications are
there? Vll Preface viii Special emphasis is placed on motivation.
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