# Normed Linear Space Part 1

1. Nov 2, 2011

### bugatti79

1. The problem statement, all variables and given/known data

Consider the normed linear space $\ell_\infty \mathbb({R})$. Let $x= \frac{n-1}{n}$, $y=(1/n) and z=2^n$

a) Are x,y,z each in $\ell_\infty \mathbb({R})$

b) What is x+y? What is $2^{1/2} y?$

c) Calculate $||x||_\infty$, $||y||_\infty$, $||x+y||_\infty$ and $||2^{1/2} y ||_\infty$

For a) Does that mean are x,y,z subspaces of $\ell_\infty \mathbb({R})$?

2. Nov 2, 2011

### Staff: Mentor

No, x, y, and z are sequences. The question is asking whether they belong to $\ell_\infty \mathbb({R})$.

3. Nov 2, 2011

### bugatti79

x and y are each an element in l_∞ as they are bounded above

z is NOT an element as its not bounded above

In b) what does it mean by 'what is'. Is that asking to evaluate?

4. Nov 2, 2011

### Staff: Mentor

You could answer b by evaluating x + y, and then saying whether it is in l(R).

5. Nov 2, 2011

### bugatti79

x+y =n implies (n) =(1,2,3,4...) which is not in l∞(R) sicne it is not bounded above.

2^(1/2)*y= 2^(n+1/2) which is not in l∞(R) either.......?

6. Nov 2, 2011

### Staff: Mentor

Yes, but here's how to say the first part better. x + y = {n} = {1, 2, 3, ...}
No, 21/2y = 21/2{1/n}. That means 21/2 times each element of the sequence {1/n}.

7. Nov 2, 2011

### bugatti79

Thanks..that was a mistake on my part. So hence it IS in l∞(R)

For c) do I attempt to evaluate each?

8. Nov 2, 2011

### Staff: Mentor

Yes and yes.
Your textbook should have a definition of the infinity norm - if not, there's one in the link I posted in the other thread.

9. Nov 3, 2011

### I like Serena

Hmm, I get a different result for x+y...

10. Nov 3, 2011

### Staff: Mentor

Let's make that x + y = {1} = {1, 1, 1, ...}

11. Nov 8, 2011

### bugatti79

For x:

1-1/n =(0,1/2,2/3,3/4...) and is bounded.

$||1-1/n||_∞$=.........? I am not sure how to calculate this. If some one can illustrate this one, I can try the others...? I know $||1-1/n||_∞=sup |x_n|$ but dont know how to calculate |x_n|

Thanks

12. Nov 8, 2011

### Staff: Mentor

See http://en.wikipedia.org/wiki/Lp_space, in the section titled Lp spaces, starting with "One also defines the ∞-norm as ...".

I believe that this is the norm to be used in your problem.

13. Nov 8, 2011

### bugatti79

THanks Mark but how to I evalute |x_n|?

14. Nov 8, 2011

### Staff: Mentor

What is the supremum of the set {0, 1/2, 2/3, 3/4, ..., n/(n + 1), ...}?

15. Nov 8, 2011

### bugatti79

I think the supremum is the smallest real number that is greater than or equal to every number in the set, hence t must be 0...?

What do i do next?

Is it something along the lines of

$|1-1/n|=\sqrt{(1-1/n)^2}=0$ where n=1

Last edited: Nov 8, 2011
16. Nov 8, 2011

### Staff: Mentor

Yes.
No. 0 is smaller than, not larger than every number in the sequence except the first.

17. Nov 8, 2011

### bugatti79

Then it doesnt exist, there is no sup...?

18. Nov 8, 2011

### Staff: Mentor

Yes! So what is the smallest number that is >= each number in the sequence {0, 1/2, 2/3, 3/4, ..., n/(n+1), ...}
No, it can't be 0. This number is too small, because it's not larger than 1/2 (for example). 2589 is larger than all the numbers in the sequence, but can you find one that is not so large?

19. Nov 9, 2011

### bugatti79

1 would be greater than or equal to every number in the set....?

20. Nov 9, 2011

### Deveno

a reasonable guess, can you prove it?