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Macroeconomics Exam Review 110

Macroeconomics Exam Review 110 - Solutions for Foundations...

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3.37 For any function, continuity implies closed graph (Exercise 2.70). To show the con- verse, assume that 𝐺 = graph( 𝑓 ) is closed. 𝑋 × 𝑌 with norm ( x , y ) = max {∥ x , y ∥} is a Banach space (Exercise 1.209). Since 𝐺 is closed, 𝐺 is complete. Also, 𝐺 is a sub- space of 𝑋 × 𝑌 . Consequently, 𝐺 is a Banach space in its own right. Consider the projection : 𝐺 𝑋 defined by ( x , 𝑓 ( x )) = x . Clearly is linear, one-to-one and onto with 1 ( x ) = ( x , 𝑓 ( x )) It is also bounded since ( x , 𝑓 ( x )) = x ∥ ≤ ∥ ( x , 𝑓 ( x ) By the open mapping theorem, 1 is bounded. For every x 𝑋 𝑓 ( x ) ∥ ≤ ∥ ( x , 𝑓 ( x )) = 1 ( x ) 1 x We conclude that 𝑓 is bounded and hence continuous. 3.38 𝑓 (1) = 5, 𝑓 (2) = 7 but 𝑓 (1 + 2) = 𝑓 (3) = 9 = 𝑓 (1) + 𝑓 (2) Similarly 𝑓 (3 × 2) = 𝑓 (6) = 15 = 3 × 𝑓 (2) 3.39 Assume 𝑓 is aﬃne. Let y = 𝑓 ( 0 ) and define
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