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**Unformatted text preview: **MATH 220: PROBLEM SET 3 DUE THURSDAY, OCTOBER 15, 2009 Problem 1. Let C ( R ) be given by ( x ) = , x < 1 , 1 + x, 1 < x < , 1 x, < x < 1 , , x > 1 , so 0, ( x ) = 0 if | x | 1 and integraltext R ( x ) dx = 1. Let j ( x ) = j ( jx ), so j ( x ) = 0 if | x | 1 /j , and integraltext j ( x ) dx = 1 for all j . (1) Show that j in D ( R ) (i.e., to be pedantic, j ). (Hint: this is essentially written up in the notes on distributions!) (2) Let C c ( R ) be such that ( x ) = 1 for | x | < 1. Show that { 2 j ( ) } j =1 is not a convergent sequence in R . Use this to conclude that { 2 j } j =1 does not converge to any distribution. (3) Show that there is no continuous extension of the map Q : f mapsto f 2 on C ( R ) (so Q : C ( R ) C ( R )) to D ( R ), i.e. there is no map Q : D ( R ) D ( R ) such that Q ( f ) = Qf...

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