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Vector Space of Bounded Sequences 
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#1
Mar812, 01:29 AM

P: 660

1. The problem statement, all variables and given/known data
Consider the vector space l_infty(R) of all bounded sequences. Decide whether or not the following norms are defined on l_infty(R) . If they are, verify by axioms. If not, provide counter example. 2. Relevant equations x in l_infty(R); x=(x_n), (i)  _# defined by x_#=x_1, (ii) \ _infty defined by  _infty =sup x_n for n in N 3. The attempt at a solution (i) a) Since x is in l_infty(R), there exist k_x >0 s.t x_n <=k_x for all n in N y is in l_infty(R), there exist k_y>0 s.t y_n <=k_y for all n in N x+y_#=(x_n+y_n)_#=x_n+y_n. Now we have that a+b <=a+b, therefore <=x_n+y_n <=k_x+k_y for all n in N <=x_#+y_# b) ax_#=ax_1 =a x_1 =aA x_# Axioms c) and d) I dont know how to attempt for this space (ii)  _infty =sup x_n for n in N Let x be in l_infty(R), x=(x_1) a) x_infty>=supx_n>=0 where supx_n>0 for n in N for all x_n in l_infty(R) b) x=0 iff x=0 c) ax_infty =supax_n =a supx_n for n in N =a x_infty d) Now sure how to use the trinagle inequality for this space..? 


#2
Mar812, 01:48 PM

P: 660

Any review of this folks?



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