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Macroeconomics Exam Review 108

# Macroeconomics Exam Review 108 - Solutions for Foundations...

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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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