# Bound Sequences

## Homework Statement

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

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

## 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

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Mark44
Mentor

## Homework Statement

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

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

## 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
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.

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.

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 $\ell_\infty$.......?

Mark44
Mentor
If we let $\ell_\infty \mathbb({R})=\left \{ \vec x=(x_n), \vec y=(y_n) \in \ell_\infty \right \}$
No, how is $\ell_\infty (R)$ defined?
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 $\ell_\infty$.......?

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

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

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

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

is {xn+yn} a bounded sequence?

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

is {xn+yn} a bounded sequence?

Yes, I believe so...

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

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

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.

Mark44
Mentor
if {xn} is a bounded sequence and {yn} is a bounded sequence,

is {xn+yn} a bounded sequence?

Yes, I believe so...

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

Now, how is "bounded sequence" defined?

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.

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

Now, how is "bounded sequence" defined?

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

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

Mark44
Mentor
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

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

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

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.....

Mark44
Mentor
I am not sure...it is possible that $|\alpha x_i|>=k$...?
Sure, that's possible, but it's not very relevant.

Can't you show that |αxi| is <= some other constant?
I dont see any constraint which states that $\alpha x_i$ cannot be >= to k.....

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

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

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.

Mark44
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

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 $k \ge |x_n|, |y_n|$

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

Mark44
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.

bugatti79 said:
...based on that the vector x is closed under multiplication...
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.

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.

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

x, y ε l_∞(R)

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.

Ok. Since we know that x and y are bounded sequences then all I need to show is that

if x_n=(x_1,x_2,x-3...) and y_n=(y_1,y_2....) then x_n+y_n=(x_1+y_1, x_2+y_2.....) Hence the set l∞(R) is closed under addition.......?

Mark44
Mentor
Yes, that's really all you need to say. Vector addition and scalar multiplication in l(R) are the same as in l(R).

Here's the link to the wiki page that deveno mentioned yesterday - http://en.wikipedia.org/wiki/Bounded_sequences.

1. Homework Statement

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

This attempt is based on a similar example in (R^3, || ||)

if x_n and y_n are each bound sequences with |x_n| >=0 for n=1,2,3 and similarly for y_n, then

axiom 1: ||x_n||∞= |x_1|+|x_2|+|x_3|... >=0 THis axiom holds.

Axiom 2: ||x_n||∞=0 IFF |x_1+|x_2|+|x_3|=0, ie each x_n=0

Axiom 3: ||ax_n||∞= |ax_1|+|ax_2|+|ax_3|
= |a|(x_1+x_2+x_3)
= |a| |x_n|∞

Axiom 4: ||x_n+y_n||∞= |x_1+y_1| +| x_2+y_2|

||x_n+y_n||∞<= |x_1|+|x_2|+|y_1|+|y_2| therefore

||x_n+y_n||∞<= ||x_n||∞ + ||y_n||∞

All 4 axioms hold therefore || ||∞ is a norm on l∞(R)............?

Have I covered all axioms correctly in this? I could not find any other axiom regarding infinity...?

Mark44
Mentor
This attempt is based on a similar example in (R^3, || ||)

if x_n and y_n are each bound sequences with |x_n| >=0 for n=1,2,3 and similarly for y_n, then

axiom 1: ||x_n||∞= |x_1|+|x_2|+|x_3|... >=0 THis axiom holds.

Axiom 2: ||x_n||∞=0 IFF |x_1+|x_2|+|x_3|=0, ie each x_n=0

Axiom 3: ||ax_n||∞= |ax_1|+|ax_2|+|ax_3|
= |a|(x_1+x_2+x_3)
= |a| |x_n|∞

Axiom 4: ||x_n+y_n||∞= |x_1+y_1| +| x_2+y_2|

||x_n+y_n||∞<= |x_1|+|x_2|+|y_1|+|y_2| therefore

||x_n+y_n||∞<= ||x_n||∞ + ||y_n||∞

All 4 axioms hold therefore || ||∞ is a norm on l∞(R)............?
You haven't actually used the definition of the infinity norm (|| ||) anywhere. The norm you seem to be using is the taxicab norm, not the infinity norm. The space here is bounded sequences, so each vector x in the space is an infinite sequence, not just a point in R3.

See (again) http://en.wikipedia.org/wiki/Bounded_sequences for a description of norms in lp spaces (including l).
There are other wiki articles on Lp spaces and norms in general.

Deveno
the "infinity norm" is the supremum (least upper bound) of the absolute value of the individual terms. since our sequences are bounded, the set of absolute values of the terms forms a set of real numbers, which is bounded above (since otherwise the sequence would either be unbounded above, or below, and we aren't considering such sequences).

a set of real numbers bounded from above, has a supremum (this is an intrinsic property of real numbers).

the usual norm, for FINITE sequences of a given length, is what you would call a "2-norm", that is we consider the norm to be the square root of the sum of the squares of the terms. this corresponds to our usual notion of distance in a finite-dimensional euclidean space. but this doesn't work for an infinite number of terms, because the terms may not "decrease fast enough" (those that do are rather special, forming an l2-space).

the reason sup is used, instead of max, is because we might have a sequence like:

{(2n- 1)/2n} which is:

(1/2, 3/4, 7/8, 15/16, 31/32, 63/64,......)

it's pretty clear this sequence is bounded below by 1/2, and bounded above by 1, but there is no "maximum" term.

You haven't actually used the definition of the infinity norm (|| ||) anywhere. The norm you seem to be using is the taxicab norm, not the infinity norm. The space here is bounded sequences, so each vector x in the space is an infinite sequence, not just a point in R3.

See (again) http://en.wikipedia.org/wiki/Bounded_sequences for a description of norms in lp spaces (including l).
There are other wiki articles on Lp spaces and norms in general.

I dont know how to prove the 4 axioms for the infinity norm, there is no example in my notes...any suggestions?

Thanks