The orthogonal of a set in the Hilbert space $\ell_2$

Question

Consider the sequences Hilbert space of complex numbers $\quad \ell_2=\{x=(x_k)_{k\in \mathbb{N}^*} \quad|\quad \sum_{k=1}^{+\infty} |x_k|^2<\infty \}$ with the inner product $<x,y>=\sum_{k=1}^{+\infty} x_k\overline{y_k}$

Let $F$ be the set $F=\{x=(x_k)_{k\in \mathbb{N}^*} \in \ell_2 \quad|\quad \sum_{k=1}^{+\infty} x_k=0 \}$

What is $F^\perp$ ?

I tried to pose $f:\ell_2 \to \mathbb{C}$ st $f(x)=\sum_{k=1}^{+\infty} x_k$ but the problem is that $f$ is not well definied for instance for the harmonic sequence $(1/k)_k$

Second attempt is write $\sum_{k=1}^{+\infty} x_k=<x,y>$ st $y=(1,1,...)$ but again $y$ is not in $\ell_2$

Answer 1

Hints:

Let $y\in F^\perp$ be given. Since $(1,-1,0,0,\ldots)\in F$, we can make a statement about the relationship between $y_1$ and $y_2$ that needs to hold. Similarly, we can make statements about other pairs of components of $y$.

Using these restrictions, you can find an upper bound of the set $F^\perp$. It remains to show that this upper bound is also a lower bound, which should be the easy part.