TITLE: The orthogonal of a set in the Hilbert space $\ell_2$
QUESTION [5 upvotes]: 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 $\ell_2$

REPLY [2 votes]: 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.