We will first show that is continuous, mathematics homework help

  

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QUESTION #7 PAGE 138:
Let A be the set of all sequences sn n 1 in l 2 such that



s  1 . Take any s  sn n 1  A , we

2
n 1 n
consider the ball B  s;2 . For any s  sn n 1  B  s; 2 , we have d  s, s  2 where d is the usual



1/2
2
 

metric on l . Therefore,   n 1 sn  sn 


2
inequality, we have




n 1

sn  sn
  2
2
 2 or


n 1
 s  s    4 . But by Cauchy-Schwartz
2
n
n

s 2   n 1 sn2  2 1  1  4 . Hence, we have
n 1 n


s  1 and thus s  A . Now, for any s  A there is a neighborhood B  s;2 contained in A .
2
n 1 n
That completes the proof that A is open.
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