Macroeconomics Exam Review 121

Macroeconomics Exam Review 121 - 3.60 A Euclidean space is a

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Unformatted text preview: 3.60 A Euclidean space is a finite-dimensional normed space, which is complete (Propo- sition 1.4). 3.61 ( x , y ) = x y satisfies the requirements of Exercise 3.59 and therefore ( x y ) 2 ( x x )( y y ) Taking square roots ) ) x y ) ) x y 3.62 By definition, the inner product is a bilinear functional. To show that it is contin- uous, let be an inner product space with inner product denote by x y . Let x x and y y be sequences in . ) ) ( x ) y x y ) ) = ) ) ( x ) y ( x ) y + ( x ) y x y ) ) ) ) ( x ) y ( x ) y ) ) + ) ) ( x ) y x y ) ) ) ) ( x ) ( y y ) ) ) + ) ) ( x x ) y ) ) Applying the Cauchy-Schwartz inequality ) ) ( x ) y x y ) ) x y y + x x y Since the sequence...
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This note was uploaded on 01/16/2012 for the course ECO 2024 taught by Professor Dr.dumond during the Fall '10 term at FSU.

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