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 &lt; \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 &lt; \infty } a_{n}^{2} &lt; \infty

a) Verify that \left\langle \left\{a_{n}\right\}^{\infty}_{1}, \left\{b_{n}\right\}^{\infty}_{1} \right\rangle = \sum_{1 \leq n &lt; \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 &lt; \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