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hw7 - Mathematics Department Stanford University Math 175...

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Mathematics Department Stanford University Math 175 Homework 7 Due Thursday May 20. 1. (i) Let X and Y be Banach spaces and let B n be the open ball around 0 in X with radius n . Let M L ( X, Y ) be an onto map, that is, Range ( M ) = Y . Argue that at least one of the sets MB n contains an open ball inside. Hint: use Q4 from HW2. (ii) Prove that there exists such r > 0 that MB 1 contains ball B r (0) and conse- quently that for all c > 0 , the set MB c (0) contains the ball B cr (0) . Hint: Show that for some x 0 and B n from (i) it is true that M ( B n + x 0 ) contains an open ball around 0 . Then use the fact that B n + x 0 can be put inside a larger ball around zero. (iii) Show that for every point u in B r (0) there exists a point x in B 2 (0) such that Mx = u. Hint: Using recursion and (ii), find the vectors x j such that u - m j =1 Mx j < r/ 2 m and k x j k < 1 / 2 m - 1 . Note that x j is an absolutely summable sequence. (iv) Conclude that there is a d > 0 so that MB 1 contains B d (0) . This is the
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