This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: H be a Hilbert space, i.e., a vector space where there is deÂ±ned an inner product ( Â· , Â· ), and with respect to which H is a complete metric space (with the distance deÂ±ned by d ( x, y ) = k xy k , where the norm is deÂ±ned by k x k = ( 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 ) = k xy k , and that xy is orthogonal to V , i.e., ( xy, 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 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....
View
Full Document
 Fall '08
 Stopple,J
 Math, Topology, Metric space, RK, lim sup

Click to edit the document details