Appendix to Chapter III
313
5 - Absolutely convergent series in a Banach space
The concept of series has no meaning in a metric space unless one has a way
of defining the necessary algebraic operations, as for example in the case of
Cartesian spaces. Everything we have said about series in nO 6, 14, 15 and
18 of Chap. II extends without the least modification to those whose terms
are vectors or points of a Cartesian space. This is so in particular in the
case, by far the most important, of absolute or unconditional convergence:
convergence of the scalar series L Ilunll implies, for all i, that of the series
formed by the ith coordinates of Un since IXil ~ Ilxll for all x E JRP, so also
that of the vector series L Un·
More generally, let E be a vector space over JR or C and suppose we are
given a norm on E, Le. a map
satisfying the following conditions:
(5.1)
IIxll = 0 {::=} x = 0,
II AX II = IAI·llxll,
Ilx + yll ~ IIxll + Ilyll
where A is any scalar, real or complex according to the case. This is, for
example, the case of the length of a vector in JR n , of the norm of uniform
convergence on the space of bounded scalar functions on any set, of the norm
on a separated pre-Hilbert space (see below), etc. A vector space endowed
with such a function norm is called a normed vector space.
Putting d(x, y) = Ilx - yll defines a distance on E and allows us to apply
all the above. Since we also have a concept of addition in E we can consider
not only sequences, but also series. A series L Un of elements of E will be
said to be convergent with sum sEE if lim (Ul + ... + Un) = s, Le. if
lim lis - (Ul + ... + un)11 = o.
Writing Sn for the sum of its n first terms, a necessary condition for convergence is, as in the case of JR or C, that for all r > 0 one has d (sp, Sq) < r for
p and q sufficiently large, in other words
(5.2)
Ilup + ... + uqll < r for p and q sufficiently large.
Now the left hand side is, by the triangle inequality, bounded by the corresponding expression for the scalar series L II Un II. If the latter converges then
the given vector series will satisfy Cauchy's criterion. To deduce convergence
from this, one clearly has to assume that E is complete.
A complete normed vector space is called a Banach space, after one of the
most gifted of the many Polish mathematicians who, during the first half of
313
5 - Absolutely convergent series in a Banach space
The concept of series has no meaning in a metric space unless one has a way
of defining the necessary algebraic operations, as for example in the case of
Cartesian spaces. Everything we have said about series in nO 6, 14, 15 and
18 of Chap. II extends without the least modification to those whose terms
are vectors or points of a Cartesian space. This is so in particular in the
case, by far the most important, of absolute or unconditional convergence:
convergence of the scalar series L Ilunll implies, for all i, that of the series
formed by the ith coordinates of Un since IXil ~ Ilxll for all x E JRP, so also
that of the vector series L Un·
More generally, let E be a vector space over JR or C and suppose we are
given a norm on E, Le. a map
satisfying the following conditions:
(5.1)
IIxll = 0 {::=} x = 0,
II AX II = IAI·llxll,
Ilx + yll ~ IIxll + Ilyll
where A is any scalar, real or complex according to the case. This is, for
example, the case of the length of a vector in JR n , of the norm of uniform
convergence on the space of bounded scalar functions on any set, of the norm
on a separated pre-Hilbert space (see below), etc. A vector space endowed
with such a function norm is called a normed vector space.
Putting d(x, y) = Ilx - yll defines a distance on E and allows us to apply
all the above. Since we also have a concept of addition in E we can consider
not only sequences, but also series. A series L Un of elements of E will be
said to be convergent with sum sEE if lim (Ul + ... + Un) = s, Le. if
lim lis - (Ul + ... + un)11 = o.
Writing Sn for the sum of its n first terms, a necessary condition for convergence is, as in the case of JR or C, that for all r > 0 one has d (sp, Sq) < r for
p and q sufficiently large, in other words
(5.2)
Ilup + ... + uqll < r for p and q sufficiently large.
Now the left hand side is, by the triangle inequality, bounded by the corresponding expression for the scalar series L II Un II. If the latter converges then
the given vector series will satisfy Cauchy's criterion. To deduce convergence
from this, one clearly has to assume that E is complete.
A complete normed vector space is called a Banach space, after one of the
most gifted of the many Polish mathematicians who, during the first half of