Basic Analysis – Gently Done Topological Vector Spaces

9: The Dual Space of a Normed Space 111
Example 9.25 Let X = ℓ ∞ and, for each n ∈ N, let ℓ m : X → K be the map
ℓ m (x) = x m , where x = (x n ) ∈ ℓ ∞ . Thus ℓ m is simply the evaluation of the m th
coordinate on ℓ ∞ .
We have |ℓm (x)| = |xm | ≤ �x�∞ and so we see that ℓm ∈ X∗ 1 for each m ∈ N.
We claim that the sequence (ℓm ) m∈N in X∗ 1 has it no w∗-convergent subsequence,
despite the fact that X∗ 1 is w∗-compact. Indeed, let (ℓmk ) k∈N be any subsequence.
Then ℓ mk → ℓ in the w∗ -topology if and only if ℓ mk
(x) → ℓ(x) in K for every
x ∈ X = ℓ ∞ . Let z be the particular element of X = ℓ ∞ given by z = (z n ) where
�
1, n = m2j , j ∈ N
zn =
.
−1, otherwise
Then ℓ mk (z) = 1 if k is even, and is equal to −1 if k is odd. So (ℓ mk (z)) cannot
converge in K.