The purpose of this book is to present some basic results in analysis that can
be used to solve various problems, so it may serve as a kind of toolbox. We do
not intend to give a complete panorama of the field. In particular, we concentrate
on real analysis and leave complex analysis aside; however we consider complex
functions now and then. Although we cannot claim that this volume is devoted to
applied analysis, our choice of topics is driven by our wish to present results that
can be applied to concrete problems. It is in this sense that this book could be called
“Motivated Analysis”. This expression is due to J.-P. Aubin and means that we shall
gather some results of analysis that may be useful in applications, even if the nature
of these applications is not the focus of the text. Thus, our route is not the choice of
some important problems that occur in applications, as in [22, 62, 87, 88, 101, 129,
130, 140, 178, 191, 236, 241, 245], but neither is it the panorama of deep advances
in mathematical concepts presented in [100] and [101].
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