1{l
f>reface
the brightest, mathematics is merely a lift to the upper strata of societyl.
My goal is not to help bright young people to arrive among the first few
in the entry competition for the French Ecole poly technique so as to find
themselves thirty years later at the service or at the head of a public or private
enterprise producing possibly war planes, missiles, military electronics, or
nuclear weapons 2 , or who will devise all kinds of financial stunts to make
their company grow beyond what they can control, and who, in both cases,
will make at least twenty times as much money as the winner of a Fields
Medal does.
The sole aim of this book thus is mathematical analysis as it was and as it
has become. The fundamental ideas which anyone must know - convergence,
continuity, elementary functions, integrals, asymptotics, Fourier series and integrals - are the subject of the first two volumes. Volume II also deals with
that part (Weierstrass) of the classical theory of analytic functions which can
be explained with the use of Fourier series, while the other part (Cauchy) will
be found at the beginning of Volume III. I have not hesitated to introduce,
sometimes very early, subjects considered as relatively advanced when they
can be explained without technical complications: series indexed by arbitrary
countable sets, the definition and elementary properties of Radon measures
in lR or
those who may care to read it at this early stage, analytic functions, the construction of Weierstrass elliptic functions as a beautiful and useful example
of a sophisticated series, etc.
I have tried to give the reader an idea of the axiomatic construction of set
theory while hoping that he will take Chap. I for what it is: a contribution to
his mathematical culture aiming at showing that the whole of mathematics
can, in principle, be built from a small number of axioms and definitions. But
a full understanding of this Chapter is not an obligatory prerequisite to an
apprenticeship in analysis. The only thing the reader will have to retain is
the naive version of set theory - standard operations on sets and functions
to which, anyway, he will get used by merely reading the next chapters -
as well as the fact that, even at the simplest level, mathematics rests upon
proofs of statements, an old art which, in French high schools and probably
elsewhere as well, is in the process of becoming obsolete because, we are told,
learning to use formulae is much more useful to most people, or because it is
too difficult for the many children of the lower strata of society (I was one in
the 1930s) who now flood the high schools ...
1 In XIX th century Cambridge, the winners of the Math Tripos would far more
often become judges or bishops than scientists.
2 One of the brightest students I have known in thirty-five years is today the head
of a holding company that controls, among other things, a chain of supermarkets. He sells Camembert, shrink-wrapped meat, Tampax, orange juice, noodles,
mustard, etc. If you have to choose, this is a more civilised way to squander your
grey matter.
f>reface
the brightest, mathematics is merely a lift to the upper strata of societyl.
My goal is not to help bright young people to arrive among the first few
in the entry competition for the French Ecole poly technique so as to find
themselves thirty years later at the service or at the head of a public or private
enterprise producing possibly war planes, missiles, military electronics, or
nuclear weapons 2 , or who will devise all kinds of financial stunts to make
their company grow beyond what they can control, and who, in both cases,
will make at least twenty times as much money as the winner of a Fields
Medal does.
The sole aim of this book thus is mathematical analysis as it was and as it
has become. The fundamental ideas which anyone must know - convergence,
continuity, elementary functions, integrals, asymptotics, Fourier series and integrals - are the subject of the first two volumes. Volume II also deals with
that part (Weierstrass) of the classical theory of analytic functions which can
be explained with the use of Fourier series, while the other part (Cauchy) will
be found at the beginning of Volume III. I have not hesitated to introduce,
sometimes very early, subjects considered as relatively advanced when they
can be explained without technical complications: series indexed by arbitrary
countable sets, the definition and elementary properties of Radon measures
in lR or
those who may care to read it at this early stage, analytic functions, the construction of Weierstrass elliptic functions as a beautiful and useful example
of a sophisticated series, etc.
I have tried to give the reader an idea of the axiomatic construction of set
theory while hoping that he will take Chap. I for what it is: a contribution to
his mathematical culture aiming at showing that the whole of mathematics
can, in principle, be built from a small number of axioms and definitions. But
a full understanding of this Chapter is not an obligatory prerequisite to an
apprenticeship in analysis. The only thing the reader will have to retain is
the naive version of set theory - standard operations on sets and functions
to which, anyway, he will get used by merely reading the next chapters -
as well as the fact that, even at the simplest level, mathematics rests upon
proofs of statements, an old art which, in French high schools and probably
elsewhere as well, is in the process of becoming obsolete because, we are told,
learning to use formulae is much more useful to most people, or because it is
too difficult for the many children of the lower strata of society (I was one in
the 1930s) who now flood the high schools ...
1 In XIX th century Cambridge, the winners of the Math Tripos would far more
often become judges or bishops than scientists.
2 One of the brightest students I have known in thirty-five years is today the head
of a holding company that controls, among other things, a chain of supermarkets. He sells Camembert, shrink-wrapped meat, Tampax, orange juice, noodles,
mustard, etc. If you have to choose, this is a more civilised way to squander your
grey matter.