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Unformatted text preview: H be a Hilbert space, i.e., a vector space where there is de±ned an inner product ( · , · ), and with respect to which H is a complete metric space (with the distance de±ned by d ( x, y ) = k xy k , where the norm is de±ned by k x k = ( x, x ) 1 / 2 ). Let V ≤ H be a closed vector subspace of H , and x ∈ H \ V . Prove that there exists a unique y ∈ V such that d ( x, V ) = k xy k , and that xy is orthogonal to V , i.e., ( xy, z ) = 0 ∀ z ∈ V. 1 2 Hint: To prove the existence of y , start by generating a sequence { y n } ⊂ V such that d ( x, y n ) ≤ d ( x, V ) + 1 /n . Prove that ∀ z, w ∈ H k wz k 2 = 2 ( k w k 2 + k z k 2 ) k w + z k 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 118a taught by Professor Stopple,j during the Fall '08 term at UCSB.
 Fall '08
 Stopple,J
 Math

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