Basic Analysis – Gently Done Topological Vector Spaces

9. The Dual Space of a Normed Space
Let X be a normed space over K. The space of all bounded linear functionals
on X, B(X,K), is denoted by X ∗ and called the dual space of X. Since K is
complete, X ∗ is a Banach space.
The Hahn-Banach theorem assures us that X ∗ is non-trivial—indeed, X ∗ separates
the points of X. Now, X ∗ is a normed space in its own right, so we may
consider its dual, X ∗∗ , this is called the bidual or double dual of X.
Let x ∈ X, and consider the mapping
ℓ ∈ X ∗ : ℓ ↦→ ℓ(x).
Evidently, this is a linear map from X ∗ into the field of scalars K. Moreover,
|ℓ(x)| ≤ �ℓ��x�, for every ℓ ∈ X ∗ ,
so we see that this is a bounded linear map from X ∗ into K, that is, it defines an
element of X ∗∗ . It turns out, in fact, that this leads to an isometric embedding of
X into X ∗∗ , as we now show.
Theorem 9.1 Let X be a normed space, and for x ∈ X, let ϕ x : X ∗ → K be
the evaluation map ϕ x (ℓ) = ℓ(x), ℓ ∈ X ∗ . Then x ↦→ ϕ x is an isometric linear
mapping of X into X ∗∗ .
Proof We have seen that ϕ x ∈ X ∗∗ for each x ∈ X. It is easy to see that x ↦→ ϕ x
is linear. We have
for all x,y ∈ X, and t ∈ K. Also
ϕ tx+y (ℓ) = ℓ(tx+y) = tℓ(x)+ℓ(y)
= tϕ x (ℓ)+ϕ y (ℓ)
|ϕ x (ℓ)| = |ℓ(x)| ≤ �ℓ��x�, for all ℓ ∈ X ∗ ,
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