Measure, Integration & Real Analysis, 2021a

166 Chapter 6 Banach Spaces
6.40 Definition infinite sum in a normed vector space
Suppose g 1 , g 2 ,...is a sequence in a normed vector space V. Then ∑ ∞ k=1 g k is
defined by
∞
∑
k=1
g k = lim n→∞
n
∑
k=1
g k
if this limit exists, in which case the infinite series is said to converge.
Recall from your calculus course that if a 1 , a 2 ,...is a sequence of real numbers
such that ∑ ∞ k=1 |a k| < ∞, then ∑ ∞ k=1 a k converges. The next result states that the
analogous property for normed vector spaces characterizes Banach spaces.
6.41
(
)
∑ ∞ k=1 ‖g k‖ < ∞ =⇒ ∑ ∞ k=1 g k converges
⇐⇒ Banach space
Suppose V is a normed vector space. Then V is a Banach space if and only if
∑ ∞ k=1 g k converges for every sequence g 1 , g 2 ,...in V such that ∑ ∞ k=1 ‖g k‖ < ∞.
Proof First suppose V is a Banach space. Suppose g 1 , g 2 ,...is a sequence in V such
that ∑ ∞ k=1 ‖g k‖ < ∞. Suppose ε > 0. Let n ∈ Z + be such that ∑ ∞ m=n‖g m ‖ < ε.
For j ∈ Z + , let f j denote the partial sum defined by
If k > j ≥ n, then
f j = g 1 + ···+ g j .
‖ f k − f j ‖ = ‖g j+1 + ···+ g k ‖
≤‖g j+1 ‖ + ···+ ‖g k ‖
∞
≤ ∑ ‖g m ‖
m=n
< ε.
Thus f 1 , f 2 ,...is a Cauchy sequence in V. Because V is a Banach space, we conclude
that f 1 , f 2 ,...converges to some element of V, which is precisely what it means for
∑ ∞ k=1 g k to converge, completing one direction of the proof.
To prove the other direction, suppose ∑ ∞ k=1 g k converges for every sequence
g 1 , g 2 ,...in V such that ∑ ∞ k=1 ‖g k‖ < ∞. Suppose f 1 , f 2 ,...is a Cauchy sequence
in V. We want to prove that f 1 , f 2 ,...converges to some element of V. It suffices to
show that some subsequence of f 1 , f 2 ,...converges (by Exercise 14 in Section 6A).
Dropping to a subsequence (but not relabeling) and setting f 0 = 0, we can assume
that
∞
‖ f k − f k−1 ‖ < ∞.
∑
k=1
Hence ∑ ∞ k=1 ( f k − f k−1 ) converges. The partial sum of this series after n terms is f n .
Thus lim n→∞ f n exists, completing the proof.
Measure, Integration & Real Analysis, by Sheldon Axler