Basic Analysis – Gently Done Topological Vector Spaces

5: Vector Spaces 51
We can extend this result to the complex case, but of course one has to take
the modulus of the various quantities appearing in the inequalities. We also need
to know that p behaves well with regard to the extraction of complex scalars from
its argument. The appropriate notion turns out to be that of a seminorm—this
will play a major rôle in the study of topological vector spaces.
Definition 5.21 Let X be a vector space over K. A mapping p : X → R is said to
be a seminorm if
(1) p(x) ≥ 0 for all x ∈ X,
(2) p(x+y) ≤ p(x)+p(y) for all x,y ∈ X,
(3) p(λx) = |λ|p(x) for all x ∈ X and all λ ∈ K.
In words, p is positive, subadditive and absolutely homogeneous. If p has the
additional property that p(x) = 0 implies that x = 0, then p is a norm on X.
The following complex version of the Hahn-Banach theorem will be seen to be
a consequence of the real version by exploiting the relationship between a complex
linear functional and its real part.
Theorem 5.22 (Hahn-Banach theorem (complex version)) Suppose that M is
a linear subspace of a vector space X over K, p is a seminorm on X, and f is a
linear functional on M such that
|f(x)| ≤ p(x) for x ∈ M.
Then there is a linear functional Λ on X that satisfies Λ ↾ M = f and
|Λ(x)| ≤ p(x) for all x ∈ X.
Proof If K = R, then, by the previous theorem, there is a suitable Λ with
−p(−x) ≤ Λ(x) ≤ p(x) for x ∈ X.
However, p(−x) = p(x) since p is a seminorm, and so |Λ(x)| ≤ p(x) for x ∈ X, as
required.
NowsupposethatK = C. ConsiderX asarealvectorspace, andnotethatRef
is a real linear functional on X satisfying Ref(x) ≤ |f(x)| ≤ p(x), for x ∈ M. By
the earlier version of the Hahn-Banach theorem, there is a real linear functional
g : X → R on X such that g ↾ M = Ref and g(x) ≤ p(x) for x ∈ X. Set
Λ(x) = g(x)−ig(ix) for x ∈ X. Then Λ : X → C is a complex linear functional.
Moreover, for any x ∈ M, we have
Λ(x) = g(x)−ig(ix) = Ref(x)−iRef(ix) = f(x)