3.189
Let
x
∈
𝐶
. Since
𝐶
is a cone,
𝜆
x
∈
𝐶
for every
𝜆
≥
0 and therefore
𝑓
(
𝜆
x
)
≥
𝑐
or
𝑓
(
x
)
≥
𝑐/𝜆
for every
𝜆
≥
0
Taking the limit as
𝜆
→ ∞
implies that
𝑓
(
x
)
≥
0
for every
x
∈
𝐶
3.190
First note that
0
∈
𝑍
and therefore
𝑓
(
0
) = 0
≤
𝑐
so that
𝑐
≥
0. Suppose that
there exists some
z
∈
𝑍
for which
𝑓
(
z
) =
𝜖
∕
= 0. By linearity, this implies
𝑓
(
2
𝑐
𝜖
z
) =
2
𝑐
𝜖
𝑓
(
z
) = 2
𝑐 > 𝑐
which contradicts the requirement
𝑓
(
z
)
≤
𝑐
for every
z
∈
𝑍
3.191
By Corollary 3.2.1, there exists
𝑓
∈
𝑋
∗
such that
𝑓
(
z
)
≤
𝑐
≤
𝑓
(
x
)
for every
x
∈
𝑆,
z
∈
𝑍
By Exercise 3.190
𝑓
(
z
) = 0
for every
z
∈
𝑍
and therefore
𝑓
(
x
)
≥
0
for every
x
∈
𝑆
Therefore
𝑍
is contained in the hyperplane
𝐻
𝑓
(0) which separates
𝑆
from
𝑍
.
3.192
Combining Theorem 3.2 and Corollary 3.2.1, there exists a hyperplane
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 Fall '10
 Dr.DuMond
 Macroeconomics, All rights reserved, Functional, Topological vector space, Linear functional

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