Basic Analysis – Gently Done Topological Vector Spaces

106 Basic Analysis
Theorem 9.13 For 1 ≤ p < ∞ the space ℓ p is separable, but the space ℓ ∞ is
non-separable.
Proof Let S denote the set of sequences of complex numbers (z n ) such that (z n ) is
eventually zero (i.e., z n = 0 for all sufficiently large n, depending on the sequence)
and such that z n has rational real and imaginary parts, for all n. Then S is a
countable set and it is straightforward to verify that S is dense in each ℓ p , for
1 ≤ p < ∞.
Note that S is also a subset of ℓ ∞ , but it is not a dense subset. Indeed, if x
denotes that element of ℓ ∞ all of whose terms are equal to 1, then �x−ζ� ∞ ≥ 1
for any ζ ∈ S.
To show that ℓ ∞ is not separable, consider the subset A of elements whose
components consist of the numbers 0,1,...,9. Then A is uncountable and the
distance between any two distinct elements of A is at least 1. It follows that the
balls {{x : �x−a� ∞ < 1/2} : a ∈ A} are pairwise disjoint. Now, if B is any dense
subset of ℓ ∞ , each ball will contain an element of B, and these will all be distinct.
It follows that B must be uncountable.
Remark 9.14 The example of ℓ 1 shows that a separable Banach space need not
have a separable dual—we have seen that the dual of ℓ 1 is ℓ ∞ , which is not
separable. This also shows that ℓ 1 is not reflexive. Indeed, this would require
that ℓ 1 be isometrically isomorphic to the dual of ℓ ∞ . Since ℓ 1 is separable, an
application of the earlier theorem would lead to the false conclusion that ℓ ∞ is
separable.
Suppose that X is a normed space with dual space X ∗ . Then, in particular,
these are both topological spaces with respect to the topologies induced by their
norms. We wish to consider topologies different from these norm topologies.
Definition 9.15 The σ(X,X ∗ )-topology on a normed space X is called the weak
topology on X.
Thus, the weak topology is the weakest topology with respect to which all
bounded linear functionals are continuous. This topology is a locally convex
topology determined by the family of seminorms P = {|ℓ| : ℓ ∈ X ∗ }. Since
X ∗ separates points of X, a normed space X is a separated topological vector
space when equipped with the weak topology. A net (x ν ) in X converges to x
with respect to the weak topology if and only if ℓ(x ν ) → ℓ(x) for each ℓ ∈ X ∗ .
Since X ∗ is a linear space, it follows immediately from Corollary 7.9, that any
linear functional on X is continuous with respect to the weak topology on X if and
only if it is a member of X ∗ , i.e., X ∗ is precisely the set of all weakly continuous
linear forms on X. Put another way, this says that every weakly continuous linear
Department of Mathematics King’s College, London