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Unformatted text preview: inner product ( Â· , Â· ), and with respect to which H is a complete metric space (with the distance deÂ±ned by d ( x,y ) = b x âˆ’ y b , where the norm is deÂ±ned by b x b = ( x,x ) 1 / 2 ). Let V â‰¤ H be a closed vector subspace of H , and x âˆˆ H \ V . Prove that there exists a unique y âˆˆ V such that d ( x,V ) = b x âˆ’ y b , and that x âˆ’ y is orthogonal to V , i.e., ( x âˆ’ y,z ) = 0 âˆ€ z âˆˆ V. 1 2 Hint: To prove the existence of y , start by generating a sequence { y n } âŠ‚ V such that d ( x,y n ) â‰¤ d ( x,V ) + 1 /n . Prove that âˆ€ z,w âˆˆ H b w âˆ’ z b 2 = 2 ( b w b 2 + b z b 2 ) âˆ’ b w + z b 2 . Use this inequality to prove that { y n } is a Cauchy sequence....
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This note was uploaded on 12/27/2011 for the course MATH 118c taught by Professor Garcia during the Fall '09 term at UCSB.
 Fall '09
 Garcia
 Math

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