220
III - Convergence: Continuous variables
8 - Series of continuous functions. Normal convergence
The concept of uniform convergence applies to series of functions when we
consider their partial sums: if s(x) = Eun(x), then uniform convergence
means, by definition, that for pEN given and for all n sufficiently large, the
partial sum sn(x) = Ul(X) + ... + un(x) is equal to s(x) to within lO-p for
all x E E, the set on which the functions x 1--+ un(x) are defined; this also
means that the "remainder" rn(x) of the series converges uniformly to o.
A particularly simple case is that of a normally convergent series of functions un(x), a concept used freely by Weierstrass, though Remmert, Funktionentheorie 1, p. 84, attributes the explicit definition and name to Baire,
Le90ns sur les theories genemle de l 'analyse, which I have not reread for more
than half a century, and whose details I seem to have forgotten: one assumes
that there exists a convergent series with positive terms E Vn whose terms
are independent of x and such that
(8.1)
lun(x)1 ~ Vn for all n E N and x E E.
This definition can be expressed in a much more striking way. Using the
notation un(x), the preceding relation is equivalent, just by the definition of
the uniform norm, to
(8.2)
So normal convergence implies
(8.3)
and in fact is equivalent to it, since we have lun(x)1 ~ lIunllE for all x E E, so
that the series with general term Vn = IlunllE satisfies the original definition.
The advantage of (3) is that it does not involve any particular series E V n ,
but the original definition is often more convenient in practice.
This said, it is first of all clear that the series E Un (x) converges absolutely
for all x. On the other hand,
Is(x) - sn(x)1 = IUn+l(X) +···1:::; vn+1 + ... ,
and since the difference between the total sum and the nth partial sum of the
series E Vn is < r for n > N, we deduce that the same holds for Is(x) -sn(x)1
for any x E E; whence uniform convergence of sn(x) to s(x), with
(8.4)
Note also the relation
(8.5)
which follows from the fact that, for all x,
III - Convergence: Continuous variables
8 - Series of continuous functions. Normal convergence
The concept of uniform convergence applies to series of functions when we
consider their partial sums: if s(x) = Eun(x), then uniform convergence
means, by definition, that for pEN given and for all n sufficiently large, the
partial sum sn(x) = Ul(X) + ... + un(x) is equal to s(x) to within lO-p for
all x E E, the set on which the functions x 1--+ un(x) are defined; this also
means that the "remainder" rn(x) of the series converges uniformly to o.
A particularly simple case is that of a normally convergent series of functions un(x), a concept used freely by Weierstrass, though Remmert, Funktionentheorie 1, p. 84, attributes the explicit definition and name to Baire,
Le90ns sur les theories genemle de l 'analyse, which I have not reread for more
than half a century, and whose details I seem to have forgotten: one assumes
that there exists a convergent series with positive terms E Vn whose terms
are independent of x and such that
(8.1)
lun(x)1 ~ Vn for all n E N and x E E.
This definition can be expressed in a much more striking way. Using the
notation un(x), the preceding relation is equivalent, just by the definition of
the uniform norm, to
(8.2)
So normal convergence implies
(8.3)
and in fact is equivalent to it, since we have lun(x)1 ~ lIunllE for all x E E, so
that the series with general term Vn = IlunllE satisfies the original definition.
The advantage of (3) is that it does not involve any particular series E V n ,
but the original definition is often more convenient in practice.
This said, it is first of all clear that the series E Un (x) converges absolutely
for all x. On the other hand,
Is(x) - sn(x)1 = IUn+l(X) +···1:::; vn+1 + ... ,
and since the difference between the total sum and the nth partial sum of the
series E Vn is < r for n > N, we deduce that the same holds for Is(x) -sn(x)1
for any x E E; whence uniform convergence of sn(x) to s(x), with
(8.4)
Note also the relation
(8.5)
which follows from the fact that, for all x,