𝐵𝐿
(
𝑋, 𝑌
)
is complete with this norm
Let (
𝑓
𝑛
) be a Cauchy sequence in
𝐵𝐿
(
𝑋, 𝑌
).
For every
x
∈
𝑋
∥
𝑓
𝑛
(
x
)
−
𝑓
𝑚
(
x
)
∥ ≤ ∥
𝑓
𝑛
−
𝑓
𝑚
∥ ∥
x
∥
Therefore (
𝑓
𝑛
(
x
)) is a Cauchy sequence in
𝑌
, which converges since
𝑌
is complete.
Define the function
𝑓
:
𝑋
→
𝑌
by
𝑓
(
x
) = lim
𝑛
→∞
𝑓
𝑛
(
x
).
𝑓
is linear since
𝑓
(
x
1
+
x
2
) = lim
𝑓
𝑛
(
x
1
+
x
2
) = lim
𝑓
𝑛
(
x
1
) + lim
𝑓
𝑛
(
x
2
) =
𝑓
(
x
1
) +
𝑓
(
x
2
)
and
𝑓
(
𝛼
x
) = lim
𝑓
𝑛
(
𝛼
x
) =
𝛼
lim
𝑓
𝑛
(
x
) =
𝛼𝑓
(
x
)
To show that
𝑓
is bounded, we observe that
∥
𝑓
(
x
)
∥
=
lim
𝑛
𝑓
𝑛
(
x
)
= lim
𝑛
∥
𝑓
𝑛
(
x
)
∥ ≤
sup
𝑛
∥
𝑓
𝑛
(
x
)
∥ ≤
sup
𝑛
∥
𝑓
𝑛
∥ ∥
x
∥
Since (
𝑓
𝑛
) is a Cauchy sequence, (
𝑓
𝑛
) is bounded (Exercise 1.100), that is there
exists
𝑀
such that
∥
𝑓
𝑛
∥ ≤
𝑀
. This implies
∥
𝑓
(
x
)
∥ ≤
sup
𝑛
∥
𝑓
𝑛
∥ ∥
x
∥ ≤
𝑀
∥
x
∥
Thus,
𝑓
is bounded.
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 Fall '10
 Dr.DuMond
 Macroeconomics, Topology, Cauchy sequence, Michael Carter, Foundations of Mathematical Economics

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