270
III - Convergence: Continuous variables
The right hand side is independent of x E K and the series E v(n), taken
over the n considered, converges, since its general term is equivalent to l/n 2 .
Whence normal convergence, qed. (This argument, modified a little, would
in fact show that (6) converges normally on any compact subset of C not
containing any n E IE).
Theorem 19 can be stated in terms of primitive functions. Consider a
sequence of functions fn on the interval I which converges uniformly to a
limit f(x) on every compact K c I. Suppose that each fn has a primitive, Le.
that there exists a differentiable function Fn on I such that F~(x) = fn(x)
for all x. We can then apply Theorem 19 to the Fn so long as the sequence
(Fn) converges for at least one point a E I, as is the case, if, for example,
Fn(a) = 0 for all n. We see then that Fn converges uniformly on all compact
K c I to a limit F, that F is differentiable, and that F' = f. In other words,
F is a primitive of f. We shall return to this result in more detail in Chap. V
a propos the "fundamental theorem of the differential and integral calculus" ,
i.e. of the relation
F(x) - F(a) = l
x f(t)dt
between a function f and its primitives.
Theorem 19 has another important corollary:
Theorem 20. Let (fn) be a sequence of functions of class CP (p::; +00) on
an interval I c R Suppose that the functions fn and all their derivatives of
order::; p converge uniformly on every compact K c I. Then the limit of the
fn is of class CP in I and
(17.7)
for all k ::; p, where f(x) = limfn(x).
To see this we need only apply Theorem 19 repeatedly, first to the fn,
whence existence of f' with (7) for k = 1, then to the f~, whence existence
of (f'Y = f" with (7) for k = 2, etc.
Theorem 19 allows one to solve all sorts of more or less classical problems. Theorem 20 for p = +00 (indefinitely differentiable functions), and
its generalisations to several variables, are the foundation of the theory of
distributions of Laurent Schwartz (Chap. V, nO 34).
18 - Extensions to unconditional convergence
In so far as they concern only absolutely convergent series, all the results of
this chapter about series of functions extend to the unconditional convergence
III - Convergence: Continuous variables
The right hand side is independent of x E K and the series E v(n), taken
over the n considered, converges, since its general term is equivalent to l/n 2 .
Whence normal convergence, qed. (This argument, modified a little, would
in fact show that (6) converges normally on any compact subset of C not
containing any n E IE).
Theorem 19 can be stated in terms of primitive functions. Consider a
sequence of functions fn on the interval I which converges uniformly to a
limit f(x) on every compact K c I. Suppose that each fn has a primitive, Le.
that there exists a differentiable function Fn on I such that F~(x) = fn(x)
for all x. We can then apply Theorem 19 to the Fn so long as the sequence
(Fn) converges for at least one point a E I, as is the case, if, for example,
Fn(a) = 0 for all n. We see then that Fn converges uniformly on all compact
K c I to a limit F, that F is differentiable, and that F' = f. In other words,
F is a primitive of f. We shall return to this result in more detail in Chap. V
a propos the "fundamental theorem of the differential and integral calculus" ,
i.e. of the relation
F(x) - F(a) = l
x f(t)dt
between a function f and its primitives.
Theorem 19 has another important corollary:
Theorem 20. Let (fn) be a sequence of functions of class CP (p::; +00) on
an interval I c R Suppose that the functions fn and all their derivatives of
order::; p converge uniformly on every compact K c I. Then the limit of the
fn is of class CP in I and
(17.7)
for all k ::; p, where f(x) = limfn(x).
To see this we need only apply Theorem 19 repeatedly, first to the fn,
whence existence of f' with (7) for k = 1, then to the f~, whence existence
of (f'Y = f" with (7) for k = 2, etc.
Theorem 19 allows one to solve all sorts of more or less classical problems. Theorem 20 for p = +00 (indefinitely differentiable functions), and
its generalisations to several variables, are the foundation of the theory of
distributions of Laurent Schwartz (Chap. V, nO 34).
18 - Extensions to unconditional convergence
In so far as they concern only absolutely convergent series, all the results of
this chapter about series of functions extend to the unconditional convergence