trying to find a inner product.

Question

Let $ \mathcal{H} $ be a hilbertspace, and let $(e_k)_{k \in \mathbb{N}}$ be a ortonormal basis for the hilbertspace. Let a x,y $ \in \mathcal{H}$ and set: $$ <x,e_k> = \frac{1}{2^k} \qquad <y,e_k> =(-1)^k \frac{1}{2^k}$$ for all k $\in \mathbb{N}$

Calculated $\mid \mid x \mid \mid, \mid \mid y \mid \mid, <x,y> $

I calculated $\mid \mid x \mid \mid, \mid \mid y \mid \mid $ both to $ \sqrt{\frac{1}{3}}$ But my problem is the last one.

I was trying to use parcivals identity, but then i need $<e_k,y>$ i can't se how to find that. Can anybody help to how to tackle this?

Answer 1

We have

$<x,y>=\sum_{k=1}^\infty<x,e_k> \overline{<y,e_k>}$,

since $x=\sum_{k=1}^\infty<x,e_k>e_k$ and $y=\sum_{k=1}^\infty<y,e_k>e_k$

FRED