Preface
VII
The sequel, in Volumes III and IV, explains subjects which require either
a much higher level of abstraction (short introductions to differential varieties and Riemann surfaces, general integration, Hilbert spaces, general harmonic analysis), or, in the last Chap. XII, a much higher level in computation
techniques: Dirichlet series of number theory, elliptic and modular functions,
connection with Lie groups. While the choice of material in Volumes I to
IV represents a coherent and nearly selfcontained block of mathematics, it
constitutes nothing more than one particular view of analysis. Other authors
could have chosen other views and, for instance, tried to lead their readers
into the theory of partial differential equations. I have not even treated differential equations in one variable: one can learn all about them in a myriad
of books, and the classical results of the theory, direct applications of the
general principles of analysis, should pose no serious problem to the student
who has assimilated these reasonably well.
In the two first volumes - Volumes III and IV are written in a much
more orthodox fashion - I have firmly emphasised, sometimes with the
aid of out of fashion excurses in ordinary language, the ideas at the basis of
analysis, and, in some cases, their historical evolution. I am not, far from it,
an expert in the history of mathematics; some mathematicians, sensing their
end coming, devote themselves to it late in life; others, younger, consider the
subject sufficiently interesting to devote a substantial part of their activity
to it; they perform a most useful task even from the pedagogical point of
view 3 since, at twenty, which I once was, one thinks only of forging ahead
without looking behind, and almost always without knowing where one is
going: where and when will one learn? I have myself preferred for a quarter
of a century to take an interest in a kind of history - science, technology, and
armaments in the XXth century - for which mathematics does not prepare
one, though there are some indirect connections. Nevertheless I have made
some effort to convey to the reader that the ideas and the techniques have
evolved, and that it took between one and two centuries for the intuitions of
the Founding Fathers to be transformed into perfectly clear concepts founded
on unassailable arguments, awaiting the great generalisations of the XXth
century.
Adopting this point of view has led me, in these first two volumes, systematically to eschew a perfectly linear exposition, organised like a clockwork
and only presenting to the reader the dominant or a la mode point of view,
with assorted Blitzbeweise, lightning proofs in the sense in which we speak of
Blitzkrieg 4 : one ratifies the result but does not comprehend the strategy until
six months after the battle. At the cost of proving the same classical results
several times over I have tried to present several methods of arguing to the
3 E. Hairer and G. Wanner, Analysis by Its History (Springer-New York, 1996), is
a prime example.
4 Rene Etiemble, a great French specialist of comparative literature, once made
a study of the styles prevailing in various kinds of activities. He came to the
conclusion that mathematical style was the closest there was to the military.
VII
The sequel, in Volumes III and IV, explains subjects which require either
a much higher level of abstraction (short introductions to differential varieties and Riemann surfaces, general integration, Hilbert spaces, general harmonic analysis), or, in the last Chap. XII, a much higher level in computation
techniques: Dirichlet series of number theory, elliptic and modular functions,
connection with Lie groups. While the choice of material in Volumes I to
IV represents a coherent and nearly selfcontained block of mathematics, it
constitutes nothing more than one particular view of analysis. Other authors
could have chosen other views and, for instance, tried to lead their readers
into the theory of partial differential equations. I have not even treated differential equations in one variable: one can learn all about them in a myriad
of books, and the classical results of the theory, direct applications of the
general principles of analysis, should pose no serious problem to the student
who has assimilated these reasonably well.
In the two first volumes - Volumes III and IV are written in a much
more orthodox fashion - I have firmly emphasised, sometimes with the
aid of out of fashion excurses in ordinary language, the ideas at the basis of
analysis, and, in some cases, their historical evolution. I am not, far from it,
an expert in the history of mathematics; some mathematicians, sensing their
end coming, devote themselves to it late in life; others, younger, consider the
subject sufficiently interesting to devote a substantial part of their activity
to it; they perform a most useful task even from the pedagogical point of
view 3 since, at twenty, which I once was, one thinks only of forging ahead
without looking behind, and almost always without knowing where one is
going: where and when will one learn? I have myself preferred for a quarter
of a century to take an interest in a kind of history - science, technology, and
armaments in the XXth century - for which mathematics does not prepare
one, though there are some indirect connections. Nevertheless I have made
some effort to convey to the reader that the ideas and the techniques have
evolved, and that it took between one and two centuries for the intuitions of
the Founding Fathers to be transformed into perfectly clear concepts founded
on unassailable arguments, awaiting the great generalisations of the XXth
century.
Adopting this point of view has led me, in these first two volumes, systematically to eschew a perfectly linear exposition, organised like a clockwork
and only presenting to the reader the dominant or a la mode point of view,
with assorted Blitzbeweise, lightning proofs in the sense in which we speak of
Blitzkrieg 4 : one ratifies the result but does not comprehend the strategy until
six months after the battle. At the cost of proving the same classical results
several times over I have tried to present several methods of arguing to the
3 E. Hairer and G. Wanner, Analysis by Its History (Springer-New York, 1996), is
a prime example.
4 Rene Etiemble, a great French specialist of comparative literature, once made
a study of the styles prevailing in various kinds of activities. He came to the
conclusion that mathematical style was the closest there was to the military.