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One of the sources of the classical differential calculus is the search for minimum
or maximum points of a real-valued function. Similarly, nonsmooth
analysis originates in extremum problems with nondifferentiable data. By now,
a broad spectrum of refined concepts and methods modeled on the theory of
differentiation has been developed.
The idea underlying the presentation of the material in this book is to start
with simple problems treating them with simple methods, gradually passing
to more difficult problems which need more sophisticated methods. In this
sense, we pass from convex functionals via locally Lipschitz continuous functionals
to general lower semicontinuous functionals. The book does not aim
at being comprehensive but it presents a rather broad spectrum of important
and applicable results of nonsmooth analysis in normed vector spaces. Each
chapter ends with references to the literature and with various exercises.
The book grew out of a graduate course that I repeatedly held at the Technische
Universität Dresden. Susanne Walther and Konrad Groh, participants
of one of the courses, pointed out misprints in an early script preceding the
book. I am particularly grateful to Heidrun P¨uhl and Hans-Peter Scheffler for
a time of prolific cooperation and to the latter also for permanent technical
support. The Institut f¨ur Analysis of the Technische Universit¨at Dresden provided
me with the facilities to write the book. I thank Quji J. Zhu for useful
discussions and two anonymous referees for valuable suggestions. I gratefully
acknowledge the kind cooperation of Springer, in particular the patient support
by Stefanie Zoeller, as well as the careful work of Nandini Loganathan,
project manager of Spi (India).
My warmest thanks go to my wife for everything not mentioned above.
Dresden, December 2006 Winfried Schirotzek
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