QuestionQuestion

Denote X = C²([0,1]) (the space of real-valued, twice continuously differentiable functions defined on [0, 1]), Y = C¹([0,1]) (the space of continuously differentiable functions on [0, 1]). Define F: (X, || . ||x) (Y, || . ||Y) by F(f) = f' + f²
(a) Suppose || ||x = Il llci and Il . ||Y = || . ||- Show that F is Fréchet differentiable at f, for any f € X. Given f, h € X, what is F'(f)h?
(b) True or False: If || . ||x = || . ||u and and || . ||Y = Il lin, then F is still differentiable at any f € X, and the formula for F' (f) h remains unchanged. No justification necessary (but no partial credit).
(c) True or False: If Il . Il X = || . ||c² and and || - ||Y = || - llc¹, then F is still differentiable at any f € X, and the formula for F' (f). h remains unchanged. No justification necessary (but no partial credit).
Note: The uniform, C¹, and C² norms are defined as follows:
Il . IIu = sup    If(x)|,
            x€[0,1]
Il . ||c¹ = IIf II u + II f' llu,
Il . ||c² = I|f|| u + IIf' IIu + IIf''II u

Suppose f : R -> C is a function such that f and that f € R ([- π,π]; C), f is π-periodic, i.e., f(t + π) = f(t) for all t € R. Prove that f(n) = 0 whenever n is an odd integer.

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