§4. Differentiable functions
271
of Chap. II; not very surprising, since this differs from classical absolute
convergence only in appearance.
By reasoning on a priori arbitrary subsets of the set of indices of the
family one can also give direct proofs which liberate one from having to
choose bijections of N onto the set of indices in question.
First, normal convergence of a series I: Ui(t) of scalar functions defined
on a set X and indexed by a set I means that there exists a family of numbers
Vi, i E I, satisfying
for all i and t. Equivalently, we may require
:L Iluilix < +00.
The series I: Ui(t) is then unconditionally convergent for any t.
Suppose further that X c tC and that, when t E X tends to a point a
adherent to X, each Ui(t) tends to a limit Ui. Then clearly again IUil s: Vi,
whence the unconditional convergence of I: Ui' If on the other hand one
denotes by sp(t) (resp. sp) the sum of Ui(t) (resp. Ui) for i E F for any
subset F of I, and the corresponding total sums by s( t) = S I (t) and s = s I,
then
Is(t) - sl = I :L[Ui(t) - uil + :L[Ui(t) - Ui11 s:
iEP
i~P
s: Isp(t) - spl + :L IUi(t) - uil s: Isp(t) - spl + 2:L Vi;
i~P
now, for all r > 0 there exists a finite F such that the last I: is < r (Chap. II,
nO 15); for this F the difference Isp(t) - spl is < r for t sufficiently close to a
(limit of a finite sum of functions). Whence Is(t) - sl < 3r and so lim set) = s.
In other words, Theorem 17 can now be written in the form
(18.1)
lim ' " Ui(t) = ' " lim Ui(t).
t-a~
~t-a
iEI
iEI
It applies in particular to normally convergent series of continuous functions.
Similarly there is a theorem on term-by-term differentiation for unconditional convergence. This time one considers, as in Theorem 19, a series
of functions Ui(X) defined and differentiable on a compact interval 38 K, assumes that the given series converges somewhere, and, finally, that the series
38 The theorems for compact convergence (Le. uniformly on all compacta) for an arbitrary interval I are no more general than the uniform convergence theorems for
a compact interval: one applies the "particular" case to each arbitrary compact
interval contained in I.
271
of Chap. II; not very surprising, since this differs from classical absolute
convergence only in appearance.
By reasoning on a priori arbitrary subsets of the set of indices of the
family one can also give direct proofs which liberate one from having to
choose bijections of N onto the set of indices in question.
First, normal convergence of a series I: Ui(t) of scalar functions defined
on a set X and indexed by a set I means that there exists a family of numbers
Vi, i E I, satisfying
for all i and t. Equivalently, we may require
:L Iluilix < +00.
The series I: Ui(t) is then unconditionally convergent for any t.
Suppose further that X c tC and that, when t E X tends to a point a
adherent to X, each Ui(t) tends to a limit Ui. Then clearly again IUil s: Vi,
whence the unconditional convergence of I: Ui' If on the other hand one
denotes by sp(t) (resp. sp) the sum of Ui(t) (resp. Ui) for i E F for any
subset F of I, and the corresponding total sums by s( t) = S I (t) and s = s I,
then
Is(t) - sl = I :L[Ui(t) - uil + :L[Ui(t) - Ui11 s:
iEP
i~P
s: Isp(t) - spl + :L IUi(t) - uil s: Isp(t) - spl + 2:L Vi;
i~P
now, for all r > 0 there exists a finite F such that the last I: is < r (Chap. II,
nO 15); for this F the difference Isp(t) - spl is < r for t sufficiently close to a
(limit of a finite sum of functions). Whence Is(t) - sl < 3r and so lim set) = s.
In other words, Theorem 17 can now be written in the form
(18.1)
lim ' " Ui(t) = ' " lim Ui(t).
t-a~
~t-a
iEI
iEI
It applies in particular to normally convergent series of continuous functions.
Similarly there is a theorem on term-by-term differentiation for unconditional convergence. This time one considers, as in Theorem 19, a series
of functions Ui(X) defined and differentiable on a compact interval 38 K, assumes that the given series converges somewhere, and, finally, that the series
38 The theorems for compact convergence (Le. uniformly on all compacta) for an arbitrary interval I are no more general than the uniform convergence theorems for
a compact interval: one applies the "particular" case to each arbitrary compact
interval contained in I.