$l^2:= \{x=(x_1,x_2,...) \ | \ \sum_{n=1}^\infty |x_n|^2

Question

$l^2:= \{x=(x_1,x_2,...) \ | \ \sum_{n=1}^\infty |x_n|^2 < \infty, \ x_n \in \mathbb{R} \ \text{for} \ n\in \mathbb{N}\}$
and $\langle x, y, \rangle := \sum_{n=1}^\infty x_ny_n$. Prove the following, 
a) $l^2$ is an inner product space.
b) $l^2$ is seperable.
c) $l^2$ is a Hilbert space.
d) find a countable orthonormal basis of $l^2$.

kinggeorge · Answer

I never was much good at this topological/analysis stuff. Especially when it came to proving things were separable. So I don't think I'll be able to help out on this one...

zzr0ck3r · Answer

npz, thanks for looking. eliassaab knows this stuff like the back of his hand, so im sure he can help out.

zzr0ck3r · Answer

@eliassaab

anonymous · Answer

To not re-invent the wheel, see this http://www.math.umn.edu/~jodeit/course/ell2.pdf

anonymous · Answer

See also http://math.stackexchange.com/questions/25885/how-to-prove-that-square-summable-sequences-form-a-hilbert-space