sa118A_1

sa118A_1 - inner product and with respect to which H is...

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

View Full Document Right Arrow Icon
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 H be a Hilbert space, i.e., a vector space where there is de±ned an
Background image of page 1

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

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: inner product ( · , · ), and with respect to which H is a complete metric space (with the distance de±ned by d ( x,y ) = b x − y b , where the norm is de±ned by b x b = ( 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 ) = b x − y b , 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 b w − z b 2 = 2 ( b w b 2 + b z b 2 ) − b w + z b 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 118c taught by Professor Garcia during the Fall '09 term at UCSB.

Page1 / 2

sa118A_1 - inner product and with respect to which H is...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online