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Macroeconomics Exam Review 59

# Macroeconomics Exam Review 59 - c 2001 Michael Carter All...

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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 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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