Question about a scalar product

Question

It is well known that in a hilbert space $H$ with orthonormal basis $(e_n)_{n=1}^{\infty }$, we have for every $f, g \in H$ $$\displaystyle{\langle f,g\rangle=\sum_{n\in\mathbb{N}}\overline{ \langle f, e_n\rangle}\langle g, e_n\rangle};$$ I would ask about the following expression $$\displaystyle{\sum_{n=1}^{\infty }\overline{\langle f, e_{n-1}\rangle}\langle f, e_{n+1}\rangle}$$ is it equal to zero or no?

Answer 1

If you take $H=\ell_2 $ and $f=\left(\frac{1}{n}\right)_{n\in\mathbb{N}}\in\ell_2$ then $$\sum_{n=2}^{\infty}\overline{\langle f, e_{n-1}\rangle }\langle f, e_{n+1}\rangle =\sum_{n=2}^{\infty}\frac{1}{n^2-1}\neq 0.$$