216
III - Convergence: Continuous variables
Yes. But before the war, in the public library at Le Havre, which was well
provided in mathematics (at 17 years old the French authors were enough
for me), his very beautiful Lefons sur les theories genemle de l'analyse (first
edition 1908), delivered at Dijon, determined my vocation. Try to find them
in the public library of your town: this might be a better test of its cultural
level that the detective stories, science fiction or the comic books for children
or "adults" that one finds in quantity, in the City of Light, in the branch of
the public library near my house.
7 - The uniform metric
For scalar sequences, the convergence of Un to u can be expressed by the relation lim d( Un, u) = O. The aim of this nO is to show that uniform convergence
of a sequence of functions can be expressed in the form lim d(fn, f) = 0, after
defining the "distance" between two functions suitably.
First of all, we shall recapitulate the definitions pertinent to least upper
bounds (Chap. II, nO 9) so as to apply them to functions with scalar values.
It is a matter of simple translation.
Let f be a function defined on an arbitrary13 set E, and with real values.
The image f(E), the set of numbers f(x) where x E E, is a subset of lR
whose least upper bound, finite or infinite, is called the least upper bound of
f over E and is written
(7.1)
sup f(x) = sup(f(E))j
xEE
this is, up to notation, the concept of the least upper bound of a family (Ui)
of real numbers (Chap. II, nO 9, end).
It is finite if and only if f is, by definition, bounded abovej it is then the
number U E lR satisfying
(SUP 1)
(SUP 2)
f(x) ::; u for any x E E,
u ::; M for any number M which majorises f,
i.e. such that f(x) ::; M for any x E E.
As in Chap. II, nO 9 one can replace (SUP 2) by
(SUP 2') for every r > 0 there exists an x such that f(x) > u - r.
It may happen that there exists an a E E where f(a) = u, in which case f
possesses an absolute maximum, but in general such a point does not exist 14 :
13 To be "any" or "arbitrary" - some humourists even speak of "arbitrary and in
other respects any" objects- is not a property, it is the total absence of any
hypothesis whatever, and in particular the hypothesis E C JR, totally irrelevant
to what follows.
14 We will, however, show in nO 9 that if f is a real function defined and continuous on a compact interval I, then there exists an a E I where f(a) attains its
maximum.
III - Convergence: Continuous variables
Yes. But before the war, in the public library at Le Havre, which was well
provided in mathematics (at 17 years old the French authors were enough
for me), his very beautiful Lefons sur les theories genemle de l'analyse (first
edition 1908), delivered at Dijon, determined my vocation. Try to find them
in the public library of your town: this might be a better test of its cultural
level that the detective stories, science fiction or the comic books for children
or "adults" that one finds in quantity, in the City of Light, in the branch of
the public library near my house.
7 - The uniform metric
For scalar sequences, the convergence of Un to u can be expressed by the relation lim d( Un, u) = O. The aim of this nO is to show that uniform convergence
of a sequence of functions can be expressed in the form lim d(fn, f) = 0, after
defining the "distance" between two functions suitably.
First of all, we shall recapitulate the definitions pertinent to least upper
bounds (Chap. II, nO 9) so as to apply them to functions with scalar values.
It is a matter of simple translation.
Let f be a function defined on an arbitrary13 set E, and with real values.
The image f(E), the set of numbers f(x) where x E E, is a subset of lR
whose least upper bound, finite or infinite, is called the least upper bound of
f over E and is written
(7.1)
sup f(x) = sup(f(E))j
xEE
this is, up to notation, the concept of the least upper bound of a family (Ui)
of real numbers (Chap. II, nO 9, end).
It is finite if and only if f is, by definition, bounded abovej it is then the
number U E lR satisfying
(SUP 1)
(SUP 2)
f(x) ::; u for any x E E,
u ::; M for any number M which majorises f,
i.e. such that f(x) ::; M for any x E E.
As in Chap. II, nO 9 one can replace (SUP 2) by
(SUP 2') for every r > 0 there exists an x such that f(x) > u - r.
It may happen that there exists an a E E where f(a) = u, in which case f
possesses an absolute maximum, but in general such a point does not exist 14 :
13 To be "any" or "arbitrary" - some humourists even speak of "arbitrary and in
other respects any" objects- is not a property, it is the total absence of any
hypothesis whatever, and in particular the hypothesis E C JR, totally irrelevant
to what follows.
14 We will, however, show in nO 9 that if f is a real function defined and continuous on a compact interval I, then there exists an a E I where f(a) attains its
maximum.