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$\begingroup$

If we have

$l^2= \{ a \in \mathbb{R}^{\mathbb{N}} | \sum_{k=0}^{\infty} |a_k|^{2} < \infty \}$ and $||a||_2 = (\sum_{k=0}^{\infty} |a_k|^2 )^{1/2}$

1

Proposition: $l^2 $ is a vector space.

We need to show that $\lambda a \in l^2$ and $(a+b)\in l^2$ for $\lambda\in \mathbb{R}$ and $a,b\in l^2$ .

for the scalar multiplication: $\lambda a = \sum _{k=0}^{\infty}|\lambda a_k|^2 = \sum_{k=0}^{\infty}|\lambda|^2|a_k|^2=|\lambda|^2\sum_{k=0}^{\infty}|a_k|^2$

the closure under addition: $a+b = \sum_{k=0}^{\infty}|a_{k}+b_{k}|^2\le \sum_{k=0}^{\infty}|a_k|^2 +2\sum_{k=0}^{\infty}|a_kb_k|+\sum_{k=0}^{\infty}|b_k|^2 $

The first and last term are finite because $a$ and $b$ are taken from $l^2$, but how do I deal with $2\sum_{k=0}^{\infty}|a_kb_k|$ ?

2

Proposition: $||.||_{2}$ is a norm on $l^2$.

We need to show that from $||x||_2 = 0$ it follows that $x=0$, that $||\lambda x||_2 = |\lambda| ||x||_2$ and $||x+y||_2 \le ||x||_2 + ||y||_2$ for $\lambda \in \mathbb{R}$ and $x,y \in l^2$.

If $||x||_2 = (\sum_{k=0}^{\infty}|x_k|^2)^{1/2} = 0 $, then $x_k = 0$ for all $k\in \mathbb{N}$.

For the scalar multiplication: $||\lambda x ||_2 = (\sum_{k=0}^{\infty}|\lambda x_k|^2 )^{1/2} = (|\lambda|^2\sum_{k=0}^{\infty}| x_k|^2 )^{1/2} = |\lambda| (\sum_{k=0}^{\infty}| x_k|^2 )^{1/2} = |\lambda| ||x||_2 $.

For the triangular inequality property: $||x+y||_2^2= \sum_{k=0}^{\infty}|x_k+y_k|^2 \le \sum_{k=0}^{\infty}|x_{k}|^2 +2\sum_{k=0}^{\infty}|x_k y_k|+\sum_{k=0}^{\infty}|y_k|^2$.

It is the same as in 1 when showing for closure under addition.... is $\sum_{k=0}^{\infty}|x_{k}y_{k}| \le (\sum_{k=0}^{\infty} | x_{k}|^2)^{1/2}(\sum_{k=0}^{\infty}|y_{k}|^2)^{1/2}$, how can one show that this inequality is true? If this was proven then:

$||x+y||_2^2= \sum_{k=0}^{\infty}|x_k+y_k|^2 \le \sum_{k=0}^{\infty}|x_{k}|^2 +2\sum_{k=0}^{\infty}|x_k y_k|+\sum_{k=0}^{\infty}|y_k|^2 \le ||x||_2^2+2||x||_2||y||_2 + ||y||_2^2 = (||x||_2+||y||_2)^2$

3

Proposition: $(l^2,||.||_2)$ is a Banach space.

We need to show completeness by showing that every Cauchy sequence converges.

If we suppose that we have elements $(a_k)_{n\in \mathbb{N}} \in l^2$ that form a Cauchy sequence and an $\epsilon > 0$ then there must be an $N$ such that : $$||a_k - a_m ||_2^2 = \sum_{n=0}^{\infty} |(a_{k})_n-(a_{m})_n|^2 < \epsilon ^2 $$ with $k,m\ge N $ from this it follows that: $$|(a_{k})_n-(a_{m})_n|< \epsilon,$$ so every subsequence is a Cauchy sequence and has a limit.

Are 1,2, and 3 correct now?

  • 0
    Your proofs of (1) and (2) are both nonsense.2012-12-06
  • 0
    Choose a sequence of epsilon so that they added up to a finite number, 1/2^n for example.2012-12-06
  • 4
    In fact, everything you've written seems to be nonsense, starting with your definition of $l^2$. You should go back to the beginning, writing down carefully what you're doing at each step.2012-12-06
  • 0
    Be careful that $a$ is a sequence of real numbers $a=(a_1, a_2, a_3,\ldots)$. Hence your Cauchy sequence $(a_n)\subset l^2$ is a sequence of real sequences $a_n=(a_{n,1},a_{n,2},\ldots,a_{n,k},\ldots)$2012-12-06
  • 2
    I don't think "proof is wrong" is a legitimate reason to down vote.2012-12-06
  • 0
    i tried to correct it... is it right now ?2012-12-06
  • 0
    Your formula $a+b = \sum_{k=0}^{\infty}|a_{k}+b_{k}|^2$ is incorrect/meaningless. The expression on the left is an element of $\ell_2$; the expression on the right is a number. Also a proof should not be just a list of formulas; add some text that explains what you are doing.2012-12-06
  • 0
    You say I should explain better what I am doing, can you give me an example from above what I could have added to show better what I have not understood ? Also, if the expression $a+b = \sum_{k=0}^{\infty}|a_{k}+b_{k}|^2$ is meaningless, then how can i show the closure and scalar multiplication?2012-12-06
  • 0
    You can write something along these lines: To prove that the sum of $a$ and $b$ belongs to $\ell_2$ for every $a,b\in \ell_2$, we show that the series $\sum_{i=1}^{\infty} (a_i + b_i)^2$ converges. 2012-12-06
  • 0
    If the main purpose of your question is to ask for checking and criticism of your proof (as opposed to asking for any proof of the mentioned fact), you should use ([tag:proof-verification]) tag to make this clear.2017-01-08

1 Answers 1