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# home2 - MATH 7338 HOMEWORK#2 Due date October 8 2009 Work...

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MATH 7338 HOMEWORK #2 Due date: October 8, 2009 Work the following problems and hand in your solutions. You may work together with other people in the class, but you must each write up your solutions independently. A subset of these will be selected for grading. Write LEGIBLY on the FRONT side of the page only, and STAPLE your pages together. 1. (a) Let δ n = (0 , 0 , . . . , 0 , 1 , 0 , 0 , . . . ) and δ 0 = (1 , 1 , 1 , . . . ) . Show that { δ n } n 0 is a Schauder basis for c. Note: You can use the fact (shown in class) that { δ n } n 1 is a Schauder basis for c 0 , so this part should be easy. (b) Show that c * is isometrically isomorphic to 1 . (c) Show that c and c 0 are topologically isomorphic and further that the two bases described above are equivalent in the sense that there exists a topological isomorphism T : c c 0 that maps the basis { δ n } n 0 for c onto the basis { δ n } n 1 for c 0 . (d) Show that if x c 0 and bardbl x bardbl = 1 , then there exist y negationslash = z c 0 with bardbl y bardbl = bardbl z bardbl = 1 such that x = ( y + z ) / 2 .

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home2 - MATH 7338 HOMEWORK#2 Due date October 8 2009 Work...

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