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Macroeconomics Exam Review 121

# Macroeconomics Exam Review 121 - c 2001 Michael Carter All...

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3.60 A Euclidean space is a finite-dimensional normed space, which is complete (Propo- sition 1.4). 3.61 𝑓 ( x , y ) = x 𝑇 y satisfies the requirements of Exercise 3.59 and therefore ( x 𝑇 y ) 2 ( x 𝑇 x )( y 𝑇 y ) Taking square roots x 𝑇 y ≤ ∥ x ∥ ∥ y 3.62 By definition, the inner product is a bilinear functional. To show that it is contin- uous, let 𝑋 be an inner product space with inner product denote by x 𝑇 y . Let x 𝑛 x and y 𝑛 y be sequences in 𝑋 . ( x 𝑛 ) 𝑇 y 𝑛 x 𝑇 y = ( x 𝑛 ) 𝑇 y 𝑛 ( x 𝑛 ) 𝑇 y + ( x 𝑛 ) 𝑇 y x 𝑇 y ( x 𝑛 ) 𝑇 y 𝑛 ( x 𝑛 ) 𝑇 y + ( x 𝑛 ) 𝑇 y x 𝑇 y ( x 𝑛 ) 𝑇 ( y 𝑛 y ) + ( x 𝑛 x ) 𝑇 y Applying the Cauchy-Schwartz inequality ( x 𝑛 ) 𝑇 y 𝑛 x 𝑇 y ≤ ∥ x 𝑛 ∥ ∥ y 𝑛 y + x 𝑛 x ∥ ∥ y Since the sequence x 𝑛 converges, it is bounded, that is there exists 𝑀 such that x 𝑛 ∥ ≤ 𝑀 for every 𝑛 . Therefore ( x 𝑛 ) 𝑇 y 𝑛 x 𝑇 y ≤ ∥ x 𝑛 ∥ ∥ y 𝑛 y + x 𝑛 x ∥ ∥ y ∥ ≤ 𝑀 y 𝑛 y + x 𝑛 x ∥ ∥ y ∥ → 0 3.63 Applying the properties of the inner product ∙ ∥ x = x 𝑇 x 0 ∙ ∥
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