Homework Help: Hilbert Space

1. Apr 26, 2013

BrainHurts

1. The problem statement, all variables and given/known data
let $\ell^{2}$ denote the space of sequences of real numbers $\left\{a_{n}\right\}^{\infty}_{1}$

such that

$\sum_{1 \leq n < \infty } a_{n}^{2} < \infty$

a) Verify that $\left\langle \left\{a_{n}\right\}^{\infty}_{1}, \left\{b_{n}\right\}^{\infty}_{1} \right\rangle = \sum_{1 \leq n < \infty } a_{n}b_{n}$ is an inner product.

b) Show that $\ell^{2}$ is a Hilbert Space.

2. Relevant equations

3. The attempt at a solution

I did part a, I believe that was easy enough, however for part b, since we're given that

$\sum_{1 \leq n < \infty } a_{n}^{2}$ = $\left\langle \left\{a_{n}\right\}^{\infty}_{1}, \left\{a_{n}\right\}^{\infty}_{1} \right\rangle$ = $\left\| \left\{a_{n}\right\}^{\infty}_{1} \right\|^{2}$ < ∞

does this mean that all sequences converge in the norm, so $\ell^{2}$ is complete and therefore a Hilbert Space?

2. Apr 26, 2013

Dick

No, it's considerably more complicated than that. You need to prove that a Cauchy sequence of sequences in $\ell^{2}$ converges to a sequence in $\ell^{2}$. I'm not an expert on this subject and if I were to try to figure out how to guide you through it, I'd probably have to look up a proof myself first. You might want to try that first. I'm kind of surprised they left this as an exercise with no other guidance.

3. Apr 26, 2013