# PS6 - zero on[0 1 5 Show that if a diﬀerentiable function...

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MATH 580: Problem Set 6 due Tuesday, October 26, 2004 1. Find the norms of the vectors ~u = (1 , 1 2 , 1 4 , 1 8 , 1 16 , ... ) and ~v = (1 , 1 3 , 1 9 , 1 27 , ... ) in the Hilbert space ` 2 using the Euclidean inner product, and verify explicitly the Schwartz inequality |h ~u,~v i| ≤ || ~u || || ~v || . 2. Given f L 2 [ a, b ], show that any orthonormal basis of L 2 deﬁnes a mapping from elements of L 2 to elements of ` 2 . 3. Keener 2.1.2 (p.93) 4. Let f ( t ) = 1 on 0 t 1. Show that the orthogonal complement of f in the space L 2 [0 , 1] is the set of all functions whose average value is
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Unformatted text preview: zero on [0 , 1]. 5. Show that if a diﬀerentiable function g is orthogonal to cos( x ) on L 2 [0 , π ], then its derivative g is orthogonal to sin( x ) on L 2 [0 , π ]. 6. Why is the space of all continuous functions ( C ) not a Banach space, for any L p norm? (a text answer will suﬃce) 7. Give two examples of a Banach space which is not a Hilbert space. 8. Keener 2.2.14 (p.97)...
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