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i IN R2, LET dr BE THE TAXICAB METRIC AND DEFINE dy (WHICH...

i IN R2, LET dr BE THE TAXICAB METRIC AND DEFINE dy (WHICH you CAN ASSUME IS A METRIC) BY (xiy's ) = mak { 1x-x', 19-9's SKETCH N1( (0,0) IN EACH OF THESE METRICS 2 a) LET ou BE A COLLECTION OF OPEN SETS, V = U u SHOW THAT V IS OPEN "EUU b) LET nEIN, U.,... Un OPEN , V = CC Uk. S...

3. Prove that, if Re Z &gt; 1, then 80 5²(2):...

3. Prove that, if Re Z &gt; 1, then 80 5²(2): = d(k) k² , k=1 for where d(k) is the number of divisors of k. 4. Prove that, if Re Z &gt; 1, then 80 o(k) 5(2)5 (z - 1) = k , k=1 where o (k) is the sum of the divisors of k. fer 6. Define the counting functio...

3. On 12, consider the right shift operator: S(X1,T2,23, ) ...

3. On 12, consider the right shift operator: S(X1,T2,23, ) = .). (a) Verify that the operator S is linear and bounded. (b) Find its adjoint S*. (c) Is S a normal operator? (a) Let I : e defined by X1 - where x = (21,202). Find T* Show that T*T = TT = 21. Find 1 and B'= 21 (c) Let T be an ...

problem (1): compact means bounded and closed for to , 1/2...

problem (1): compact means bounded and closed for to , 1/2 / f.1/1' n ..} is bounded nzl 5/1/11 n < l for -any and orake I therefore { 01 1/2 / / 3 1 } C[0011 is bounded. n - / closed because for: 1/m / n s (o,us) \ fortin's / / = " R / , R / ) , + , n nil 1/5 )...

1. [25pts] (a) 18 pts) Show 15 518 and S for , € P. (...

1. [25pts] (a) 18 pts) Show 15 518 and S for , € P. (b) [8 pts) Let (X. M. 11) be a measure space and suppose 1 <<< S < oc. Show if 1(X) < ac and 11/11/18) for f EL'(()) if (X) = 1. (c) [8 pts] On C² (with the usual inner product) let T : C² ...

12. Let X and Y be normed spaces. Show that a bounded sesqui...

12. Let X and Y be normed spaces. Show that a bounded sesquilinear form h on XXY is jointly continuous in both variables. 13. (Hermitian form) Let X be a vector space over a field K. A Hermitian sesquilinear form or, simply, Hermitian form h on X X X is a mapping h: XxX K such that for all x, y...

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