Measure, Integration & Real Analysis, 2021a

Section 6D Linear Functionals 177
Now we can prove the promised result about the existence of discontinuous linear
functionals on every infinite-dimensional normed vector space.
6.62 discontinuous linear functionals
Every infinite-dimensional normed vector space has a discontinuous linear
functional.
Proof Suppose V is an infinite-dimensional vector space. By 6.61, V has a basis
{e k } k∈Γ . Because V is infinite-dimensional, Γ is not a finite set. Thus we can assume
Z + ⊂ Γ (by relabeling a countable subset of Γ).
Define a linear functional ϕ : V → F by setting ϕ(e j ) equal to j‖e j ‖ for j ∈ Z + ,
setting ϕ(e j ) equal to 0 for j ∈ Γ \ Z + , and extending linearly. More precisely, define
a linear functional ϕ : V → F by
)
ϕ(
∑ α j e j = ∑ α j j‖e j ‖
j∈Ω
j∈Ω∩Z +
for every finite subset Ω ⊂ Γ and every family {α j } j∈Ω in F.
Because ϕ(e j )=j‖e j ‖ for each j ∈ Z + , the linear functional ϕ is unbounded,
completing the proof.
Hahn–Banach Theorem
In the last subsection, we showed that there exists a discontinuous linear functional
on each infinite-dimensional normed vector space. Now we turn our attention to the
existence of continuous linear functionals.
The existence of a nonzero continuous linear functional on each Banach space is
not obvious. For example, consider the Banach space l ∞ /c 0 , where l ∞ is the Banach
space of bounded sequences in F with
‖(a 1 , a 2 ,...)‖ ∞ = sup
k∈Z + |a k |
and c 0 is the subspace of l ∞ consisting of those sequences in F that have limit 0. The
quotient space l ∞ /c 0 is an infinite-dimensional Banach space (see Exercise 15 in
Section 6C). However, no one has ever exhibited a concrete nonzero linear functional
on the Banach space l ∞ /c 0 .
In this subsection, we show that infinite-dimensional normed vector spaces have
plenty of continuous linear functionals. We do this by showing that a bounded linear
functional on a subspace of a normed vector space can be extended to a bounded
linear functional on the whole space without increasing its norm—this result is called
the Hahn–Banach Theorem (6.69).
Completeness plays no role in this topic. Thus this subsection deals with normed
vector spaces instead of Banach spaces.
We do most of the work needed to prove the Hahn–Banach Theorem in the next
lemma, which shows that we can extend a linear functional to a subspace generated
by one additional element, without increasing the norm. This one-element-at-a-time
approach, when combined with a maximal object produced by Zorn’s Lemma, gives
us the desired extension to the full normed vector space.
Measure, Integration & Real Analysis, by Sheldon Axler