# Bound Sequences

1. Oct 31, 2011

### bugatti79

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

a) Prove that $\ell_\infty \mathbb({R})$ is a subspace of $\ell \mathbb({R})$

b) Show that $\left \| \right \|_\infty$ is a norm on $\ell_\infty (\mathbb{R})$

3. The attempt at a solution

For a) I guess we have to show that $\vec{x} + \vec{y} \in \ell_\infty \mathbb({R})$ and $\alpha \vec({x}) \in \ell_\infty \mathbb({R})$

but I dont know how to proceed....thanks

Last edited by a moderator: Oct 31, 2011
2. Oct 31, 2011

### Staff: Mentor

Start by assuming that x and y are in $\ell_\infty \mathbb({R})$, and say what it means for a vector to be in that space. Then add the vectors together. Is the sum in that space as well? Same thing for the scalar multiplication part.

3. Oct 31, 2011

### bugatti79

If we let $\ell_\infty \mathbb({R})=\left \{ \vec x=(x_n), \vec y=(y_n) \in \ell_\infty \right \}$

If x and y are vectors in $\ell_\infty$ then x+y is also in $\ell_\infty$
If $\alpha \in \mathbb{R}$ then $\alpha \vec x$ is also in $\ell_\infty$.......?

4. Oct 31, 2011

### Staff: Mentor

No, how is $\ell_\infty (R)$ defined?

5. Oct 31, 2011

### bugatti79

This is the only definition I have in my notes with the addition of inserting the vector y

$\ell_\infty \mathbb({R})=\left \{ \vec x=(x_n), \vec y=(y_n) \in \ell_\infty \mathbb({R}) \right \}$

6. Oct 31, 2011

### Deveno

if {xn} is a bounded sequence and {yn} is a bounded sequence,

is {xn+yn} a bounded sequence?

7. Oct 31, 2011

### bugatti79

Yes, I believe so...

8. Oct 31, 2011

### bugatti79

Mark asked a good question. This is something I dont have in my notes...?

9. Oct 31, 2011

### Deveno

(excerpt from wikipedia): If p = ∞, then ℓ is defined to be the space of all bounded sequences (in K, the range of the sequences).

thus ℓ(R) is the space of all bounded real sequences.

10. Oct 31, 2011

### Staff: Mentor

Which is what you need to show. Also that {axn} is a bounded sequence.

Now, how is "bounded sequence" defined?

11. Oct 31, 2011

### Deveno

Let $\ell_\infty \mathbb({R})$ be the set of bounded real sequences with k > 0 such that $\left | x_n \right |\le k$

this is from another post of yours. it has the information you need.

12. Nov 2, 2011

### bugatti79

Let $\ell_\infty \mathbb({R})$ be the set of bounded real sequences with k > 0 such that $\left | x_n \right |\le k$

if $\vec x, \vec y \in \ell_\infty \mathbb({R})$ then there exist $n_1, n_2 \in N$ such that

$\vec x = (x_1,x_2,...x_{n1},0,0...)$ and $\vec y = (y_1,y_2,...y_{n2},0,0...) \therefore \vec x +\vec y= (x_1+y_1, x_2+y_2....x_n+y_n,0,0...)$ where $k \ge |x_n|, |y_n|$

$\alpha \vec x = (\alpha x_1, \alpha x_2,....\alpha x_n,0,0..)$ where

$\alpha \in \mathbb{R}$................?

13. Nov 2, 2011

### Staff: Mentor

If every element xi in the sequence satisfies |xi| <= k, what can you say about αxi?

14. Nov 2, 2011

### bugatti79

I am not sure...it is possible that $|\alpha x_i|>=k$...?

I dont see any constraint which states that $\alpha x_i$ cannot be >= to k.....

15. Nov 2, 2011

### Staff: Mentor

Sure, that's possible, but it's not very relevant.

Can't you show that |αxi| is <= some other constant?

16. Nov 2, 2011

### bugatti79

Based on the question asked and all the info I have in #12, I dont see any other constant at play....
I wouldnt understand how or when one would look at some 'other' constant.

17. Nov 2, 2011

### Staff: Mentor

If |x| < 3, then certainly 2|x| < 6, right?

On a side note, could you ease up a bit on the LaTeX, especially for symbols that don't actually require it? These threads with lots of LaTeX take a long time to render on my browser. Many of the things that you write can be done using the Quick Symbols that appear on the right after you click Go Advanced.

Everything below is done without using LaTeX.
αxi
πr2
∫x2dx

18. Nov 2, 2011

### bugatti79

OK, thanks for the advice on LaTex.

So yes, that example makes sense....based on that the vector x is closed under multiplication. How about my attempt for addition as above? Hopefully I have a) answered.

For part b) I have to show that || ||_∞ satifies 4 specific axioms, right?

Thanks

19. Nov 2, 2011

### Staff: Mentor

In the previous post, #18, why are you working with finite sequences? Addition in l is term-by-term. All you have to show is that, if x and y are bounded sequences, then x + y is also a bounded sequence.

No, vectors aren't closed under scalar multiplication - the set that they belong to, l(R) in this case, is closed under scalar multiplication.

For the b part, verify the properties in the definition of a norm.

20. Nov 2, 2011

### bugatti79

I dont know how to write it any other way...

x, y ε l_∞(R)