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convince yourself that (C ([−1, 1]; R), ρp) is a metric space for all p ∈ N, where ρp : C ([−1, 1]; R) × C ([−1, 1]; R) → R is defined as ρp(f, g) = Z 1 −1 |f(x) − g(x)| p dx 1/p for all f, g ∈ C ([−1, 1]; R). We have shown in class that the metric space (C ([−1, 1]; R), ρ∞) is complete, where ρ∞ is the uniform distance. Show that, in contrast, the metric space (C ([−1, 1]; R), ρp) is incomplete for all p ∈ N [Hint: Consider the sequence (fk)k of functions defined for all k ∈ N by fk(x)

Exercise 8 (11 points). Following the steps in Exercise 10.2 of Homework 1, convince yourselfthat (‘6’([—1,1];R),pp) is a metric space for all p E N, where pp: ‘6’([—1, 1];R) x ‘6([—1,1];R)—> Ris deﬁned as 1 Upmtg) = (fl Iva) — germ) for an 129 e ecu—1, 1];R)- We have shown in class that the metric space (‘e’([—1,1];R),pm) is cemplete, where poo is theuniform distance. Show that, in contrast, the metric space (?([71,1];R),pp) is incomplete for all p E N [HintConsider the sequence (fig);c of functions deﬁned for all k E N by fiche) = —1 for all m E [—1, —1/k),ﬁsts) = km for all :1: E [—1/k,1/k) and fk($) = 1 for all :5 E [1/k,1].]