4. (14 points) Solve either part (a) or part (b) of this problem. If you are...
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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 ||$||2 = ( 18 (2) dx)" be the L2 norm on C[0, 1]. Also let (Z) be the space of complex sequences (In)nez with 1/2 |(n)|e= |n| - ( HP) 10. Show that C[0, 1] is not complete under the L norm, whereas (Z) is complete under the norm. (b) Let C[0, 1] be the space of complex-valued C functions on [0, 1], i.e. the space of all functions f: [0, 1] C so that f= F on [0, 1] for some function F has a continuous derivative on some open interval containing [0, 1]. Let 1/2 |||f||w, = (*|f(2)1 +15 (2)|dx) be the W1,2 norm on C[0, 1]. Also let ,2 (Z) be the space of complex sequences (En) nZ with 1/2 |(n) 2 (1+4n) |n| = ( 2 < 00. -DO Show that C[0, 1] is not complete under the W2 norm, whereas (.2(Z) is complete under the (,2 norm. (You may use, without proof, the existence of a C function y so that p(x)=0 when r < 1/2, and p(x) = 1 when r>1. You may also assume that 2 (Z) is complete.) Remark. It can be shown that there exists a norm-preserving (hence injective) linear map from (C10, 1], W.2) into .2 (Z). So the latter can be taken to be the completion of C[0, 1] under the W12 norm. (You are not required to prove this fact.) 4. (14 points) Solve either part (a) or part (b) of this problem. If you are taking MATH6110, you should solve part (b). (a) Let C[0, 1] be the space of complex-valued continuous functions on [0, 1], and let 1/2 ||$||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.)
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Income Tax Fundamentals 2013
ISBN: 9781285586618
31st Edition
Authors: Gerald E. Whittenburg, Martha Altus Buller, Steven L Gill
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