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Macroeconomics Exam Review 126

# Macroeconomics Exam Review 126 - c 2001 Michael Carter All...

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be the corresponding closest points in 𝑆 . Then ( y 1 y 2 ) ( x 1 x 2 ) 2 = ( ( y 1 y 2 ) ( x 1 x 2 ) ) 𝑇 ( ( y 1 y 1 ) ( x 1 x 2 ) ) = ( y 1 y 2 ) 𝑇 ( y 1 y 2 ) + ( x 1 x 2 ) 𝑇 ( x 1 x 2 ) 2( y 1 y 2 ) 𝑇 ( x 1 x 2 ) = y 1 y 2 2 + x 1 x 2 2 2( y 1 y 2 ) 𝑇 ( x 1 x 2 ) = y 1 y 2 2 + x 1 x 2 2 2( y 1 y 2 ) 𝑇 ( x 1 x 2 ) 2 x 1 x 2 2 + 2( x 1 x 2 ) 𝑇 ( x 1 x 2 ) = y 1 y 2 2 − ∥ x 1 x 2 2 + 2 ( ( x 1 x 2 ) ( y 1 y 2 ) ) 𝑇 ( x 1 x 2 ) = y 1 y 2 2 − ∥ x 1 x 2 2 + 2( x 1 y 1 ) 𝑇 ( x 1 x 2 ) 2( x 2 y 2 ) 𝑇 ( x 1 x 2 ) = y 1 y 2 2 − ∥ x 1 x 2 2 2( x 1 y 1 ) 𝑇 ( x 2 x 1 ) 2( x 2 y 2 ) 𝑇 ( x 1 x 2 ) so that y 1 y 2 2 − ∥ x 1 x 2 2 = ( y 1 y 2 ) ( x 1 x 2 ) 2 + 2( x 1 y 1 ) 𝑇 ( x 2 x 1 ) + 2( x 2 y 2 ) 𝑇 ( x 1 𝑥 2 ) Using Exercise 3.72 ( x 1 y 1 ) 𝑇 ( x 2 x 1 ) 0 and ( x 2 y 2 ) 𝑇 ( x 1 x 2 ) 0 which implies that the left-hand side y 1 y 2 2 − ∥ x 1 x 2 2 0 or x 1 x 2 = 𝑔 ( y 1 ) 𝑔 ( y 2 ) ∥ ≤ ∥ y 1 y 2 𝑔 is Lipschitz continuous. 3.75 Let 𝑆 = kernel 𝑓 . Then 𝑆 is a closed subspace of 𝑋 . If 𝑆 = 𝑋 , then 𝑓 is the zero functional and
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