18.2. SEPARATION THEOREMS 483

and let f be a linear map from π (X) to F defined by

f (πx)≡ z(x).

(This is well defined because if π (x) = π (x1), then yi (x) = yi (x1) for i = 1, · · · ,n and so

ρA (x− x1) = 0.

Thus,|z(x1)− z(x)|= |z(x1− x)| ≤CρA (x− x1) = 0.)

Extend f to all of Fn and denote the resulting linear map by F . Then there exists a vector

α = (α1, · · · ,αn) ∈ Fn

with α i = F (ei) such thatF (β ) = α ·β .

Hence for each x ∈ X ,

z(x) = f (πx) = F (πx) =n

∑i=1

α iyi (x)

and so

z =n

∑i=1

α iyi ∈ Y.

This proves the theorem.

18.2 Separation TheoremsIt will always be assumed that X is a locally convex topological vector space. A set, K, issaid to be convex if whenever x,y ∈ K,

λx+(1−λ )y ∈ K

for all λ ∈ [0,1].

Definition 18.2.1 Let U be an open convex set containing 0 and define

m(x)≡ inf{t > 0 : x/t ∈U}.

This is called a Minkowski functional.

Proposition 18.2.2 Let X be a locally convex topological vector space. Then m is definedon X and satisfies

m(x+ y)≤ m(x)+m(y) (18.2.4)

m(λx) = λm(x) if λ > 0. (18.2.5)

Thus, m is a gauge function on X.