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.
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
The Attempt at a Solution
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
<=k_x+k_y for all n in N
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
d) Now sure how to use the trinagle inequality for this space..?