sa1 - H be a Hilbert space, i.e., a vector space where...

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Self-Assessment Questions – Math 118A, Fall 2009 1. For any sequence { c n } of positive numbers, lim inf n →∞ c n +1 c n lim inf n →∞ n c n lim sup n →∞ n c n lim sup n →∞ c n +1 c n . 2. For any two real sequences { a n } , { b n } , prove that lim sup n →∞ ( a n + b n ) lim sup n →∞ a n + lim sup n →∞ b n . 3. Let ( X, d ) be a metric space. Prove that the union (intersection) of an arbitrary family of open (closed) sets is open (closed). 4. Prove that a perfect set in R k (with the Euclidean distance) is uncount- able. 5. Imitate the proof of theorem 2.43 in the book to obtain the following result: If R k = n =1 F n , where each F n is a closed subset of R k , then at least one F n has nonempty interior. An equivalent statement is the following: If G n is a dense open subset of R k for n = 1 , 2 , . . . , then n =1 G n is not empty (in fact, it is dense in mathbR k ). This result is a special case of Baire’s theorem. 6. Let
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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 x-y 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 x-y k , and that x-y is orthogonal to V , i.e., ( x-y, 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 w-z 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.

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sa1 - H be a Hilbert space, i.e., a vector space where...

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