3-2 3. APPLICATIONS OF HAHN-BANACH
To see how this could work, let’s apply the result to the Riemann integral on C[0, 1].
We conclude the existence of a positive linear functional ` : B[0, 1] → R that gives `(f) =
R
1
0
f(x)dx. This gives an “integral” of arbitrary bounded functions, without a measurability
condition. Since the integral is a linear functional, it is finitely additive. Of course it is
not countably additive since we know from real analysis that such a thing doesn’t exist.
Furthermore, the integral is not uniquely defined.
Nonetheless, the extension is also not arbitrary, and the constraint of positivity actually
pins down `(f) for many functions. For example, consider χ
(a,b)
(the characteristic function
of an open interval). By positivity we know that
f ≤ χ
(a,b)
≤ g, f, g ∈ C[0, 1] =⇒
Z
1
0
f(x)dx ≤ `(χ
(a,b)
) ≤
Z
1
0
g(x)dx.
Taking the sup over f and inf over g, using properties only of the Riemann integral, we see
that `(χ
(a,b)
) = b − a (hardly a surprising result). Likewise, we can see that `(χ
U
) = |U| for
any open set, and by finite additivity `(χ
F
) = 1 − `(F
c
) = 1 − |F
c
| = |F | for any closed set
(| · | is Lebesgue measure). Finally if S ⊂ [0, 1] and
sup {`(F ) : F closed and F ⊂ S} = inf {`(U) : U open and U ⊃ S}
then `(χ
S
) must be equal to these two numbers. We see that `(χ
S
) = |S| for any Lebesgue
measurable set. We have just painlessly constructed Lebesgue measure from the Riemann
integral without using any measure theory!
The rigidity of the extension is a bit surprising if we compare with what happens in
finite dimensions. For instance, consider the linear functional `(x, 0) = x defined on Y =
{(x, 0)} ⊂ R
2
. Let P be the cone {(x, y) : |y| ≤ αx}. So ` is P ∩ Y -positive. To extend `
to all of R
2
we need to define `(0, 1). To keep the extension positive we need only require
y`(0, 1) + 1 ≥ 0
if |y| ≤ α. Thus we must have |`(0, 1)| ≤ 1/α, and any choice in this interval will work.
Only when α = ∞ and the cone degenerates to a half space does the condition pin `(0, 1)
down. So some interesting things happen in ∞ dimensions.
A second example application assigning a limiting value to sequences
a = (a
1
, a
2
, . . .).
Let B denote the space of all bounded R-valued sequences and let L denote the subspace of
sequences with a limit. We quickly conclude that there is an positive linear extension of the
positive linear functional lim to all of B. We would like to conclude a little more, however.
After all, for convergent sequences,
lim a
n+k
= lim a
n
for any k.
To formalize this property we define a linear map on B by
T (a
1
, a
2
, . . .) = (a
2
, a
3
, . . .).
(T is the “backwards shift” operator or “left translation.”) Here,
Definition 3.2. Let X
1
, X
2
be linear spaces over a field F . A linear map T : X
1
→ X
2
is a function such that
T (x + ay) = T (x) + aT (y) ∀x, y ∈ X
1
and a ∈ F.
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