Basic Analysis – Gently Done Topological Vector Spaces

50 Basic Analysis
and so, multiplying by t and using the linearity of f and the positive homogeneity
of p, we get
f(x)−tµ ≤ p(x−tx 1 ) for x ∈ M and t > 0.
Thus,
f 1 (x+sx 1 ) ≤ p(x+sx 1 ) for x ∈ M and any s < 0.
From (2), replacing y by t −1 y, with t > 0, we get
Thus, for any t > 0,
i.e.,
f(t −1 y)+µ ≤ p(t −1 y +x 1 ), y ∈ M.
f(y)+tµ ≤ p(y +tx 1 ), y ∈ M,
f 1 (y +tx 1 ) ≤ p(y +tx 1 ), for any y ∈ M, and t > 0.
This inequality also holds for t = 0, by hypothesis on f. Combining these inequalities
shows that
f 1 ≤ p on M 1 .
We have shown that f can be extended from M to a subspace with one extra
dimension, whilst retaining the bound in terms of p. To obtain an extension to
the whole of X, we shall use Zorn’s lemma. (An idea would be to apply the above
procedure again and again, but one then has the problem of showing that this
would eventually exhaust the whole of X. Use of Zorn’s lemma is a way around
this difficulty.) Let E denote the family of ordered pairs (N,h), where N is a
linear subspace of X containing M and h is a real linear functional on N such
that h ↾ M = f and h(x) ≤ p(x) for all x ∈ N. The pair (M,f) (and also (M1 ,f1 )
constructed above) is an element of E, so that E is not empty. Moreover, E is
partially ordered by extension, that is, we define (N,h) � (N ′ ,h ′ ) if N ⊆ N ′ and
h ′ ↾ N = h. Let C be any totally ordered subset of E and set N ′ = �
(N,h)∈CN. Let x ∈ N ′ . Suppose that x ∈ N1∩N 2 , where (N1 ,h1 ) and (N2 ,h2 ) are members
of C. Then either (N1 ,h1 ) � (N2 ,h2 ) or (N2 ,h2 ) � (N1 ,h1 ). In any event,
h1 (x) = h2 (x). Hence we may define h ′ on N ′ by the assignment h ′ (x) = h(x) if
x ∈ N with (N,h) ∈ C. Then h ′ is a linear functional on N ′ such that h ′ ↾ M = f
and h ′ (x) ≤ p(x) for all x ∈ N ′ . This means that (N ′ ,h ′ ) is an upper bound for
C in E. By Zorn’s lemma, E contains a maximal element (N,Λ), say. If N �= X,
then we could construct an extension of (N,Λ), as earlier, which would contradict
maximality. We conclude that N = X and that Λ is a linear functional Λ : X → R
such that Λ ↾ M = f, and Λ(x) ≤ p(x) for all x ∈ X.
Replacingxby−xgivesΛ(−x) ≤ p(−x), i.e.,−Λ(x) ≤ p(−x)andso−p(−x) ≤
Λ(x) for x ∈ X.
Department of Mathematics King’s College, London