𝑥
∈
𝑋
and therefore
𝑓
+
𝑔
∈
𝐵
(
𝑋
). Hence,
𝐵
(
𝑋
) is closed under addition and
scalar multiplication; it is a subspace of the linear space
𝐹
(
𝑋
). We conclude that
𝐵
(
𝑋
) is a normed linear space.
3. To show that
𝐵
(
𝑋
) is complete, assume that (
𝑓
𝑛
) is a Cauchy sequence in
𝐵
(
𝑋
).
For every
𝑥
∈
𝑋
∣
𝑓
𝑛
(
𝑥
)
−
𝑓
𝑚
(
𝑥
)
∣ ≤ ∥
𝑓
𝑛
−
𝑓
𝑚
∥ →
0
Therefore, for
𝑥
∈
𝑋
,
𝑓
𝑛
(
𝑥
) is a Cauchy sequence of real numbers. Since
ℜ
is
complete, this sequence converges. Define the function
𝑓
(
𝑥
) = lim
𝑛
→∞
𝑓
𝑛
(
𝑥
)
We need to show
∙ ∥
𝑓
𝑛
−
𝑓
∥ →
0 and
∙
𝑓
∈
𝐵
(
𝑋
)
(
𝑓
𝑛
) is a Cauchy sequence. For given
𝜖 >
0, choose
𝑁
such that
∥
𝑓
𝑛
−
𝑓
𝑚
∥
< 𝜖/
2
for very
𝑚, 𝑛
≥
𝑁
. For any
𝑥
∈
𝑋
and
𝑛
≥
𝑁
,
∣
𝑓
𝑛
(
𝑥
)
−
𝑓
(
𝑥
)
∣ ≤ ∣
𝑓
𝑛
(
𝑥
)
−
𝑓
𝑚
(
𝑥
)
∣
+
∣
𝑓
𝑚
(
𝑥
)
−
𝑓
(
𝑥
)
∣
≤ ∥
𝑓
𝑛
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 Fall '10
 Dr.DuMond
 Macroeconomics, Vector Space, Cauchy sequence, Michael Carter, Foundations of Mathematical Economics

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