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?
     
Know someone interested in this topic? Share this thead via email, Google+, Twitter, or Facebook

Have something to add?