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Macroeconomics Exam Review 44

# Macroeconomics Exam Review 44 - Solutions for Foundations...

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1.212 Let 𝑆 be an open set according to the ∥⋅∥ 𝑎 and let x 0 be a point in 𝑆 . Since 𝑆 is open, it contains an open ball in the ∥⋅∥ 𝑎 topology about x 0 , namely 𝐵 𝑎 ( x 0 , 𝑟 ) = { x 𝑋 : x x 0 𝑎 < 𝑟 } ⊆ 𝑆 Let 𝐵 𝑏 ( x 0 , 𝑟 ) = { x 𝑋 : x x 0 𝑏 < 𝑟 } be the open ball about x 0 in the ∥⋅∥ 𝑏 topology. The condition (1.15) implies that 𝐵 𝑏 ( x 0 , 𝑟 ) 𝐵 𝑎 ( x 0 , 𝑟 ) 𝑆 and therefore x 0 𝐵 𝑏 ( x 0 , 𝑟 ) 𝑆 𝑆 is open in the ∥⋅∥ 𝑏 topology. Similarly, any 𝑆 open in the ∥⋅∥ 𝑏 topology is open in the ∥⋅∥ 𝑎 topology. 1.213 Let 𝑋 be a normed linear space of dimension 𝑛 . and let { x 1 , x 2 , . . . , x 𝑛 } be a basis for 𝑋 . Let ∥⋅∥ 𝑎 and ∥⋅∥ 𝑏 be two norms on 𝑋 . Every x 𝑋 has a unique representation x = 𝛼 1 x 1 + 𝛼 2 x 2 + ⋅ ⋅ ⋅ + 𝛼 𝑛 x 𝑛 Repeated application of the triangle inequality gives x 𝑎 = 𝛼 1 x 1 + 𝛼 2 x 2 + ⋅ ⋅ ⋅ + 𝛼 𝑛 x 𝑛 𝑎 𝑛 𝑖 =1 𝛼 𝑖 x 𝑖 𝑎 = 𝑛 𝑖 =1 𝛼 𝑖 ∣ ∥ x 𝑖 𝑎 𝑘 𝑛 𝑖 =1 𝛼 𝑖 where
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