Mathematics Department Stanford UniversityMath 175 Homework 7Due Thursday May 20.1. (i) LetXandYbe Banach spaces and letBnbe the open ball around0inXwith radiusn. LetM∈L(X, Y)be an onto map, that is,Range(M) =Y.Argue that at least one of the setsMBncontains an open ball inside.Hint: use Q4 from HW2.(ii) Prove that there exists suchr >0thatMB1contains ballBr(0)and conse-quently that for allc >0,the setMBc(0)contains the ballBcr(0).Hint: Show that for somex0andBnfrom (i) it is true thatM(Bn+x0)containsan open ball around0.Then use the fact thatBn+x0can be put inside a largerball around zero.(iii) Show that for every pointuinBr(0)there exists a pointxinB2(0)suchthatMx=u.Hint: Using recursion and (ii), find the vectorsxjsuch thatu-∑mj=1Mxj<r/2mandkxjk<1/2m-1.Note thatxjis an absolutely summable sequence.(iv) Conclude that there is ad >0so thatMB1containsBd(0). This is the
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