Register to reply

Vector Space of Bounded Sequences

by bugatti79
Tags: bounded, sequences, space, vector
Share this thread:
Mar8-12, 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

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

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..?
Phys.Org News Partner Science news on
'Smart material' chin strap harvests energy from chewing
King Richard III died painfully on battlefield
Capturing ancient Maya sites from both a rat's and a 'bat's eye view'
Mar8-12, 01:48 PM
P: 660
Any review of this folks?

Register to reply

Related Discussions
Bounded sequences and convergent subsequences in metric spaces Calculus 5
Cauchy Sequences are Bounded Calculus & Beyond Homework 3
Proving bounded monotonic sequences must converge Calculus & Beyond Homework 1
Proof of the collections of sequences are linear spaces or vector space. Calculus & Beyond Homework 6
Bounded sequences Calculus & Beyond Homework 4