Measure, Integration & Real Analysis, 2021a

Section 6D Linear Functionals 183
14 Show that there exists a linear functional ϕ : l ∞ → F such that
for all (a 1 , a 2 ,...) ∈ l ∞ and
|ϕ(a 1 , a 2 ,...)| ≤‖(a 1 , a 2 ,...)‖ ∞
ϕ(a 1 , a 2 ,...)= lim
k→∞
a k
for all (a 1 , a 2 ,...) ∈ l ∞ such that the limit above on the right exists.
15 Suppose B is an open ball in a normed vector space V such that 0/∈ B. Prove
that there exists ϕ ∈ V ′ such that
for all f ∈ B.
Re ϕ( f ) > 0
16 Show that the dual space of each infinite-dimensional normed vector space is
infinite-dimensional.
A normed vector space is called separable if it has a countable subset whose closure
equals the whole space.
17 Suppose V is a separable normed vector space. Explain how the Hahn–Banach
Theorem (6.69) for V can be proved without using any results (such as Zorn’s
Lemma) that depend upon the Axiom of Choice.
18 Suppose V is a normed vector space such that the dual space V ′ is a separable
Banach space. Prove that V is separable.
19 Prove that the dual of the Banach space C([0, 1]) is not separable; here the norm
on C([0, 1]) is defined by ‖ f ‖ = sup| f |.
[0, 1]
The double dual space of a normed vector space is defined to be the dual space of
the dual space. If V is a normed vector space, then the double dual space of V is
denoted by V ′′ ; thus V ′′ =(V ′ ) ′ . The norm on V ′′ is defined to be the norm it
receives as the dual space of V ′ .
20 Define Φ : V → V ′′ by
(Φ f )(ϕ) =ϕ( f )
for f ∈ V and ϕ ∈ V ′ . Show that ‖Φ f ‖ = ‖ f ‖ for every f ∈ V.
[The map Φ defined above is called the canonical isometry of V into V ′′ .]
21 Suppose V is an infinite-dimensional normed vector space. Show that there is a
convex subset U of V such that U = V and such that the complement V \ U is
also a convex subset of V with V \ U = V.
[See 8.25 for the definition of a convex set. This exercise should stretch your
geometric intuition because this behavior cannot happen in finite dimensions.]
Measure, Integration & Real Analysis, by Sheldon Axler