Basic Analysis – Gently Done Topological Vector Spaces

80 Basic Analysis
Hence, for x ∈ B,
p i (x/s) ≤ d
s
< r whenever s > d/r,
i.e., x/s ∈ V(0,p 1 ,...,p n ;r) ⊆ U, whenever s > d/r. It follows that
B ⊆ sV(0,p 1 ,...,p n ;r) ⊆ sU for s > d/r
and we conclude that B is bounded, as required.
Next we consider a further version of the Hahn-Banach theorem.
Theorem 7.27 (Hahn-Banach theorem) Let X be a locally convex topological
vector space determined by the family P of seminorms. Suppose that M is a linear
subspace of X and that λ : M → K is a continuous linear functional on M. Then
there is a constant C > 0 and a finite set of elements p 1 ,...,p n in P, such that
|λ(x)| ≤ C(p 1 (x)+···+p n (x)) for x ∈ M.
Furthermore, there is a continuous linear functional Λ on X such that
and Λ ↾ M = λ.
|Λ(x)| ≤ C(p 1 (x)+···+p n (x)) for x ∈ X,
Proof The relative topology on M is determined by the restrictions of the seminorms
in P to M. Indeed, a neighbourhood base at 0 in M is given by the sets
V(0,p 1 ,...,p k ;r)∩M = {x ∈ M : p i (x) < r, 1 ≤ i ≤ k}
= {x ∈ M : (p i ↾ M)(x) < r, 1 ≤ i ≤ k}
= V(0,p 1 ↾ M,...,p k ↾ M;r).
Let P M denote the collection of restrictions P M = {p ↾ M : p ∈ P}. Since
λ : M → K is a continuous linear functional on M, it follows that there is C > 0
and p 1 ↾ M,...,p n ↾ M in P M such that
|λ(x)| ≤ C(p 1 ↾ M(x)+···+p n ↾ M(x)), x ∈ M,
which is precisely the statement that
|λ(x)| ≤ C(p 1 (x)+···+p n (x)), x ∈ M.
For x ∈ X, set q(x) = C(p 1 (x)+···+p n (x)). Then q is a seminorm on X and
we have |λ(x)| ≤ q(x) on M. By our earlier version of the Hahn-Banach theorem,
Theorem 5.22, there is a linear functional Λ on X such that Λ ↾ M = λ and
|Λ(x)| ≤ q(x) for x ∈ X.
This bound implies that Λ is continuous, by Corollary 7.9.
Department of Mathematics King’s College, London