Consider $H=\lbrace x\in l^2:\Sigma_{k\geq1}k^2|x_k|^2<\infty\rbrace$. I have this norm $\|·\|_H:H\to\mathbb{R}$ given by $\|x\|_H=(\Sigma_{k\geq1}k^2|x_k|^2)^{\frac{1}{2}}$. And I need to calculate its inner product.
I used parallelogram law and calculated its inner product as $\langle x,y\rangle=\Sigma_{k\geq1}k^2x_ky_k$. I'm not sure whether I calculated this correct. Does the inner product should be $\langle x,y\rangle=\Sigma_{k\geq1}k^2|x_ky_k|$? What is the correct inner product?