Orthonormal zero Function

Question

I have this exercise

Let H be a Hilbert space with orthonormal basis $\{e_n | n\in N\}$ and let $f_n = e_n + e_{n+1}$

If $\langle f,f_n \rangle = 0$ for all $n$ how do I prove that $f=0$

I think i can do it for a even n for an odd one i cant... Can anyone help?

Answer 1

You can also directly consider the Fourier coefficients of the basis. The assumption says $ ⟨f,e_{n+1}⟩=- ⟨f,e_n⟩$, so $$⟨f,e_n⟩=(-1)^{n-1}⟨f,e_1⟩$$ and per the answer of T. Bongers, the asymptotic behavior following from Parseval $$ \|f\|^2=\sum_{n=1}^\infty |⟨f,e_n⟩|^2 $$ demands that all coefficients are zero.

Answer 2

Here's a sketch of one technique: Since $f \perp e_1 + e_2$ and $f \perp e_2 + e_3$, it's immediate that $f \perp e_1 - e_3$. Repeat to conclude that $f \perp e_1 + e_4$, and eventually that

$$f \perp e_1 + (-1)^n e_n$$

Now use a general result about Hilbert spaces to show that $\langle f, e_n \rangle \to 0$ as $n \to \infty$, so that $f \perp e_1$. Repeat the argument for $e_2$, and so on.