Measure, Integration & Real Analysis, 2021a

Section 6C Normed Vector Spaces 165
From now on, all metric space notions in the context of a normed vector space
should be interpreted with respect to the metric introduced in the previous result.
However, usually there is no need to introduce the metric d explicitly—just use the
norm of the difference of two elements. For example, suppose (V, ‖·‖) is a normed
vector space, f 1 , f 2 ,... is a sequence in V, and f ∈ V. Then in the context of a
normed vector space, the definition of limit (6.8) becomes the following statement:
lim
k→∞ f k = f means lim
k→∞
‖ f k − f ‖ = 0.
As another example, in the context of a normed vector space, the definition of a
Cauchy sequence (6.12) becomes the following statement:
A sequence f 1 , f 2 ,...in a normed vector space (V, ‖·‖) is a Cauchy sequence
if for every ε > 0, there exists n ∈ Z + such that ‖ f j − f k ‖ < ε for
all integers j ≥ n and k ≥ n.
Every sequence in a normed vector space that has a limit is a Cauchy sequence
(see 6.13). Normed vector spaces that satisfy the converse have a special name.
6.37 Definition Banach space
A complete normed vector space is called a Banach space.
In other words, a normed vector space
V is a Banach space if every Cauchy sequence
in V converges to some element
of V.
The verifications of the assertions in
Examples 6.38 and 6.39 below are left to
the reader as exercises.
In a slight abuse of terminology, we
often refer to a normed vector space
V without mentioning the norm ‖·‖.
When that happens, you should
assume that a norm ‖·‖ lurks nearby,
even if it is not explicitly displayed.
6.38 Example Banach spaces
• The vector space C([0, 1]) with the norm defined by ‖ f ‖ = sup| f | is a Banach
space.
[0, 1]
• The vector space l 1 with the norm defined by ‖(a 1 , a 2 ,...)‖ 1 = ∑ ∞ k=1 |a k| is a
Banach space.
6.39 Example not a Banach space
• The vector space C([0, 1]) with the norm defined by ‖ f ‖ = ∫ 1
0
| f | is not a
Banach space.
• The vector space l 1 with the norm defined by ‖(a 1 , a 2 ,...)‖ ∞ = sup
k∈Z + |a k | is
not a Banach space.
Measure, Integration & Real Analysis, by Sheldon Axler