The aim of this book is to present the basic facts of linear functional anal-
ysis related to applications to some fundamental aspects of mathematical
If mathematics is supposed to show common general facts and struc-
tures of particular results, functional analysis does this while dealing with
classical problems, many of them related to ordinary and partial differential
equations, integral equations, harmonic analysis, function theory, and the
calculus of variations.
In functional analysis, individual functions satisfying specific equations
are replaced by classes of functions and transforms which are determined by
each particular problem. The objects of functional analysis are spaces and
operators acting between them which, after systematic studies intertwining
linear and topological or metric structures, appear to be behind classical
problems in a kind of cleaning process.
In order to make the scope of functional analysis clearer, I have chosen
to sacrifice generality for the sake of an easier understanding of its methods,
and to show how they clarify what is essential in analytical problems. I
have tried to avoid the introduction of cold abstractions and unnecessary
terminology in further developments and, when choosing the different topics,
I have included some applications that connect functional analysis with other
The text is based on a graduate course taught at the Universitat de
Barcelona, with some additions, mainly to make it more self-contained. The
material in the first chapters could be adapted as an introductory course
on functional analysis, aiming to present the role of duality in analysis, and
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