# Prove that the little l(p) sequence spaces are complete.

• Sep 30th 2010, 06:28 PM
davismj
Prove that the little l(p) sequence spaces are complete.
http://i55.tinypic.com/2u8b6ts.jpg

The book gave me the "hint" to mimic the proof of the l(1) space, but I can't find it anywhere. I have no idea how to approach this. I understand I want to show that every cauchy space converges to something in l(p), or that every sequence of sequences becomes arbitrarily close to another sequence. I know it follows from completeness on R, but thats all I know. I'm totally lost.
• Sep 30th 2010, 06:49 PM
Iondor
||x_n - x_m|| >= ||x_n_k - x_m_k||

So, if (x_n) is a Cauchy sequence, then so are (x_n_k) for all k.
Since R is complete every (x_n_k) converges against a number y_k.
Now show that (x_n) converges against y:=(y_1,y_2,...) in the p-norm.
• Sep 30th 2010, 08:17 PM
davismj
Disagree.

$\|x\| = \big(\sum_{n=1}^{\infty}|x_n|^p\big)^{1/p}$

So if $x_n = \{x_{n_k}\}_{k=1}^{\infty}$ and m similarly, then $\|x_n - x_m\|$ may be greater than, less than, or equal to $\|x_{n_k} - x_{m_k}\|$

Also, I'm pretty sure my way is the wrong way to approach this. I don't think we're interested in subsequences of $x_n$ at all