19.2.2.10 Convergence in Normed Spaces

Condition: show that the norms \ (\ | x \ | _ {c} = \ max _ {0 \ leq t \ leq 1} | x (t) | \) and \ (\ | x \ | _ {c_ {1}} = \ int_ {0} {t} | x (t) | d t \ quadad \) not \ (\ quad \) equivalents \ (\ quad \) in space \ (c [0.1] \). Indication: Examine the sequence \ (x_ {n} (t) = t {n} \) for convergence in these norms.