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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

## 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.

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online