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1. Denote by Cb,even (R;R) the set of all even, bounded and contin- uous functions u : R R. Prove that Cb,even(R;R) is a subspace of Cb(R;R), and that the set, {pk: : R R/Px(x)==22, E E Nu {0} } is a linearly independent subset of Cb,even(R;R) which is not a Hamel basis. 2. Using integration by parts prove for, the estimate, (C. a |u(t)| ²dr) 1/2 < 2max(la1,1bl) (C. a b 1/2 Using this justify why 1/2 u (jo a defines a norm on C¹ *((a,6)) n Co ((a,6)). Hint: Start by writing, b dt = b 1 . 2 dt a a and then apply integration by parts.

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