Vector Space of Bounded Sequences

  1. 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>=sup|x_n|>=0 where sup|x_n|>-0 for n in N for all x_n in l_infty(R)
    b) ||x||=0 iff x=0
    c) ||ax||_infty =sup|ax_n|
    =|a| sup|x_n| for n in N
    =|a| ||x||_infty

    d) Now sure how to use the trinagle inequality for this space..?
     
  2. jcsd
  3. Any review of this folks?
     
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