Basic Analysis – Gently Done Topological Vector Spaces

82 Basic Analysis
Thus, for t ∈ K and u ∈ M,
That is,
|λ(tx 0 +u)| = |t| ≤ 1
r s(tx 0 +u).
|λ(x)| ≤ 1
r s(x) for x ∈ M 1
which shows that λ : M 1 → K is continuous. By the Hahn-Banach theorem,
Theorem 7.27, it follows that there is a continuous linear functional Λ on X which
extends λ (and obeys the same bound). In particular, Λ(x 0 ) = 1 and Λ(x) = 0 for
all x ∈ M.
Theorem 7.31 Let X be a vector space over K and suppose that F is a family
of linear functionals on X. Then the σ(X,F)-topology coincides with the locally
convex topology T P determined by the family P = {|ℓ| : ℓ ∈ F} of seminorms
on X. This latter topology is Hausdorff if and only if F is a separating family,
i.e., for any x ∈ X with x �= 0 there is some ℓ ∈ F such that ℓ(x) �= 0.
Proof First we observe that if ℓ is a linear functional on X then |ℓ|is a seminorm,
and the condition that F be a separating family of linear functionals is equivalent
to P being a separating family of seminorms.
To show that the vector space topology determined by P is the same as the
σ(X,F)-topology, we shall show that they have the same convergent nets. To this
end, let (x ν ) be a net in X. We have
x ν → x with respect to the σ(X,F)-topology,
⇐⇒ ℓ(x ν ) → ℓ(x), for each ℓ ∈ F,
⇐⇒ ℓ(x ν −x) → 0, for each ℓ ∈ F,
⇐⇒ |ℓ(x ν −x)| → 0, for each ℓ ∈ F,
⇐⇒ x ν −x → 0 with respect to T P ,
⇐⇒ x ν → x with respect to T P .
Hence these two topologies have the same convergent nets, the same closed sets
and therefore the same open sets.
Examples 7.32
1. Equip C(R), the linear space of complex-valued continuous maps on R, with the
separating family of seminorms P = {p K : K compact in R}, where p K is given
by p K (f) = sup{|f(x)| : x ∈ K} for f ∈ C(R).
Convergence in (C(R),T P ) is uniform convergence on compact sets.
Department of Mathematics King’s College, London