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Unformatted text preview: Therefore . For every x ( ( x ) z ( z ) x ) z = ( x ) z z ( z ) x z = 0 since z . Therefore ( x ) = ( z ) z 2 x z = x ( z ( z ) z 2 ) = x y where y = z ( z ) z 2 3.76 is always complete (Proposition 3.3). To show that it is a Hilbert space, we have to that it has an inner product. For this purpose, it will be clearer if we use an alternative notation < x , y > to denote the inner product of x and y . Assume is a Hilbert space. By the Riesz representation theorem (Exercise 3.75), for every there exists y such that ( x ) = < x , y > for every x Furthermore, if y represents and y represents , then y + y represents + and y represents since ( + )( x ) = ( x ) + ( x ) = < x , y > + < x , y...
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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.
- Fall '10