Verifying Inner Product & Showing $\ell^{2}$ is a Hilbert Space

[BrainHurts]

Homework Statement

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.

Homework Equations

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?

 

[Dick]

Homework Statement

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.

Homework Equations

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?

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.

 

[BrainHurts]

hmm any suggested readings?

 

[Dick]

hmm any suggested readings?

Any number of textbooks. This popped up online. http://www.math.umn.edu/~jodeit/course/ell2.pdf