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Vilmos Komornik
Topology,Calculus and Approximation
Topology, Calculus and Approximation.pdf
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© Springer-Verlag London Ltd. 2017
For God’s sake, I beseech you, give it up.Fear it no less than sensual passions because it too may take all your time and deprive you
of your health, peace of mind and happiness in life...
Letter of Farkas Bolyai to his son,
April 4, 1820.
I created a new, different world out of nothing...
Letter of János Bolyai to his father,
November 2, 1823.
Preface
This textbook contains the lecture notes of three one-semester courses given by the
author to third year students at the University of Strasbourg. We assume that the
reader is familiar with the calculus of one real variable. The first part on Topology is
used everywherein the sequel. The following two parts on Differential calculus and
Approximation methods are logically independent.
We have made much effort to select the material covered by the lectures, to
formulate aesthetical and general statements, to seek short and elegant proofs, and
to illustrate the results with simple but pertinent examples. (See also the remarks
on p. 369.) Our work is strongly influenced by the beautiful lectures of Professors
ákos Császár and László Czách during the 1970s at the Eötvös Loránd University
in Budapest, and more generally by the mathematical tradition created by Leopold
Fejér, Frédéric Riesz, Paul Turán, Paul Erd˝ os and others.
On p. 337 we cite many papers of historical importance, indicating the origin of
most of the notions and theorems treated here. They often contain different versions
of the theorems we treat, illustrating the genesis of mathematical interest.
We suggest that, on the first reading, the reader should skip the material marked
by ?. At the end of each chapter we give some exercises. However, the most
important exercises are incorporated into the text as examples and remarks, and
the reader is expected to fill in the missing details.
We list on p. ix some books of general mathematical interest.
We thank á. Besenyei, C. Baud, L. Czách, C. Disdier, D. Dumont, J. Gerner,
P. Loreti, C.-M. Marle, P. Martinez, M. Mehrenberger, P.P. Pálfy, M. Pedicini, P.
Pilibossian, J. Saint Jean Paulin, Z. Sebestyén, A. Simonovits, L. Simon, Mrs. B.
Szénássy, G. Szigeti, J. Vancostenoble, Zs. Votisky and the editors of Springer for
their precious help.
This book is dedicated to the memory of Paul Erd˝ os.
Strasbourg, France Vilmos Komornik
March 26, 2017
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