hw10 - ` 2 norm of the sequence of terms of its Fourier...

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Math 318 HW #10 Due 3:00 PM Friday, April 29 Reading: § 32–37. Problems: 1. Exercise 34.36. Comment: The unit ball not being compact is a common feature of infinite-dimensional normed linear spaces which can be a serious nuisance. In fact, one can prove that the unit ball in a normed linear space is compact if and only if the space is finite-dimensional. 2. Exercise 38.20. Note that Parseval’s Theorem is correctly stated in this exercise, but the statement is incorrect in Corollary 37.10. Comment: If you’ve ever encountered ` 2 , Parseval’s Theorem tells us that the L 2 norm of a function is the same as the
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Unformatted text preview: ` 2 norm of the sequence of terms of its Fourier series. 3. Exercise 38.22. Note that I’ve been using h f, g i 2 in class to indicate the L 2 inner product of f and g , whereas Wilcox & Myers use f · g to mean the same thing. Also, there’s a typo in the statement of the problem: the first terms in the Fourier series for f and g should be a 2 and α 2 , and not a 2 2 and α 2 2 . Comment: This is just the L 2 analogue of the usual way we compute inner products (dot products) in R n by summing the products of the components. 4. Exercise 38.27. 1...
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