Basic Analysis – Gently Done Topological Vector Spaces

9: The Dual Space of a Normed Space 103
Theorem 9.10 For any 1 ≤ p ≤ ∞, ℓ p is a Banach space. Moreover, if 1 ≤ p < ∞,
the dual of ℓ p is ℓ q , where q is the exponent conjugate to p. Furthermore, for each
1 < p < ∞, the space ℓ p is reflexive.
Proof We have already discussed these spaces for p = 1 and p = ∞. For the rest,
it follows from the preceding results that ℓ p is a linear space and that �·� p is a
norm on ℓ p . The completeness of ℓ p , for 1 < p < ∞, follows in much the same
way as that of the proof for p = 1.
To show that ℓ p∗ = ℓ q , we use the pairing as in Hölder’s inequality. Indeed, for
any y = (y n ) ∈ ℓ q , define ψ y on ℓ p by ψ y : x = (x n ) ↦→ �
n x n y n
. Then Hölder’s
inequality implies that ψ y is a bounded linear functional on ℓ p and the subsequent
proposition (with the rôles of p and q interchanged) shows that �ψ y � = �y� q .
To show that every bounded linear functional on ℓ p has the above form, for
some y ∈ ℓ q , let λ ∈ ℓ p∗ , where 1 ≤ p < ∞. Let yn = λ(e n ), where e n =
(δ nm ) m∈N ∈ ℓ p . Then for any x = (x n ) ∈ ℓ p ,
λ(x) = λ ��
n
x n e n
� ��
= xnλ(en ) � = �
Hence, replacing x n by sgn(x n y n )x n , we see that
�
n
n
|x n y n | ≤ �λ��x� p .
n
x n y n .
For any N ∈ N, denote by y ′ the truncated sequence (y 1 ,y 2 ,...,y N ,0,0,...).
Then
�
n
|x n y ′ n | ≤ �λ��x� p
and, taking the supremum over x with �x� p = 1, we obtain the estimate
�y ′ � q ≤ �λ�.
It follows that y ∈ ℓ q (and that �y� q ≤ �λ�). But then, by definition, ψ y = λ, and
we deduce that y ↦→ ψ y is an isometric mapping onto ℓ p∗ . Thus, the association
y ↦→ ψ y is an isometric linear isomorphism between ℓ q and ℓ p∗ .
Finally, we note that the above discussion shows that ℓ p is reflexive, for all
1 < p < ∞.