4. (14 points) Solve either part (a) or part (b) of this problem. If you are taking MATH6110, you should solve part (b). (a) Let C[0, 1] be the space of complex-valued continuous functions on [0, 1], and let 1/2 ||$||2 = ( 18 (2) dx)" be the L2 norm on C[0, 1]. Also let (Z) be the space of complex sequences (In)nez with 1/2 |(n)|e= |n| - ( HP) 10. Show that C[0, 1] is not complete under the L norm, whereas (Z) is complete under the norm. (b) Let C[0, 1] be the space of complex-valued C functions on [0, 1], i.e. the space of all functions f: [0, 1] C so that f= F on [0, 1] for some function F has a continuous derivative on some open interval containing [0, 1]. Let 1/2 |||f||w, = (*|f(2)1 +15 (2)|dx) be the W1,2 norm on C[0, 1]. Also let ,2 (Z) be the space of complex sequences (En) nZ with 1/2 |(n) 2 (1+4n) |n| = ( 2 < 00. -DO Show that C[0, 1] is not complete under the W2 norm, whereas (.2(Z) is complete under the (,2 norm. (You may use, without proof, the existence of a C function y so that p(x)=0 when r < 1/2, and p(x) = 1 when r>1. You may also assume that 2 (Z) is complete.) Remark. It can be shown that there exists a norm-preserving (hence injective) linear map from (C10, 1], W.2) into .2 (Z). So the latter can be taken to be the completion of C[0, 1] under the W12 norm. (You are not required to prove this fact.)