Welcome to our community

Be a part of something great, join today!

lp space

dray

Member
Feb 5, 2012
37
Could anybody help me with the attached problem?

I've managed to prove part (ii) and (iv) of the question, but have no idea where to begin with part (i). I'm assuming that the result follows by using the definitions, but cannot make any progress with it.

For part (iii) I have thought about looking at the ration of the two norms and then using the squeeze rule but again, have yet to make any progress with this.

Any help would be much appreciated here.

Do we also know when MHF will be up and running again?
 

Attachments

Last edited:

girdav

Member
Feb 1, 2012
96
Where is the attached problem?
 

Jameson

Administrator
Staff member
Jan 26, 2012
4,035
Do we also know when MHF will be up and running again?
I don't see any attachment and please read this thread in regards to MHF. This is a new site that will remain for a long time. Shoot me a PM if you can't work out the attachment issue.
 

girdav

Member
Feb 1, 2012
96
For question i), if $a=0$ it's clear, and otherwise, for $r<\infty$ use the fact that the sequence $\{a_n\}$ converges to $0$. For $r=\infty$, it's not true (take $a_n=1-\frac 1n$).
 

dray

Member
Feb 5, 2012
37
For question i), if $a=0$ it's clear, and otherwise, for $r<\infty$ use the fact that the sequence $\{a_n\}$ converges to $0$. For $r=\infty$, it's not true (take $a_n=1-\frac 1n$).
Thanks for replying so quickly, Girdav. However, I'm still not sure how to translate this into a proof as such.
 

girdav

Member
Feb 1, 2012
96
That's your job, so I will only give a sketch of proof. If $a=0$ then the result is obvious, and if $a\neq 0$, since $\lim_{n\to \infty}a_n=0$ we can find an integer $N$ such that $|a_n|\leq \frac{\lVert a\rVert_{\infty}}2$ if $n\geq N$. So the index we are looking for is between $0$ and $N-1$.
 

dray

Member
Feb 5, 2012
37
I've tried and tried and tried and still don't get this!
 

girdav

Member
Feb 1, 2012
96
What do you get?
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,189
Do we also know when MHF will be up and running again?
Nobody knows. What I do know is that MHB is not going away any time soon. It also does not suffer from the same problem MHF did: a site owner who had orphaned the site.
 

dray

Member
Feb 5, 2012
37
What do you get?
It's just the first part I am unsure of. Am I to show that a convergent sequence attains its supremum?
 

girdav

Member
Feb 1, 2012
96
Yes, and here it's a little more easy since it converges to $0$. Did you try with my hints?
 

dray

Member
Feb 5, 2012
37
That's your job, so I will only give a sketch of proof. If $a=0$ then the result is obvious, and if $a\neq 0$, since $\lim_{n\to \infty}a_n=0$ we can find an integer $N$ such that $|a_n|\leq \frac{\lVert a\rVert_{\infty}}2$ if $n\geq N$. So the index we are looking for is between $0$ and $N-1$.
OK - I give up with this problem! I just cannot see how you make the jump from converges to zero on to attaining its supremum.

Where did the 2 in the denominator come from in your inequality?

I'm at least partially happy that I have managed to prove the remaining parts of the problem.