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Question: i consider the sequence of continuous piecewise linear functions in...

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I. Consider the sequence of continuous piecewise linear functions in C°(0, 1]) defined by 0, where the coefficients an and bn are chosen so that fn is continuous. (a) Find the values of an and bn (b) Show that this is not a Cauchy sequence with respect to the Loo norm rej0,1 (o) Show that, on the contrary, this is a Cauchy sequence with respect to the LP norm for any 1 p < oo. (d) Finally show that, considering Co (0, 1]) as a subset of the normed vector space (L (,) of all integrable functions with finite LP norm, the sequence ff.) converges to the (integrable but discontinuous) function
where the coefficients a, and b, are chosen so that f Shou ny converg e untegrable but discontinuous) function f(x) = 0, z E [0, 2)
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