Show that (C ([1, 1]; R), p) is a metric space for all p N, where p : C ([1, 1]; R) C ([1, 1]; R) R is defined as p(f, g) = Z 1 1 |f(x) g(x)| p dx 1/p for all f, g C ([1, 1]; R). Show that, the metric space (C ([1, 1]; R), p) 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 yourself that ('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)> R is dened as 1 Up mtg) = (fl Iva) germ) for an 129 e ecu1, 1];R)- We have shown in class that the metric space ('e'([1,1];R),pm) is cemplete, where poo is the uniform distance. Show that, in contrast, the metric space (?([71,1];R),pp) is incomplete for all p E N [Hint Consider the sequence (fig);c of functions dened for all k E N by fiche) = 1 for all m E [1, 1/k), sts) = km for all :1: E [1/k,1/k) and fk($) = 1 for all :5 E [1/k,1].]