Decompose $x$ into $F$ and $F^{\perp }$ in $l^2(\mathbb{N}, \mathbb{C})$

Question

Let $l^2(\mathbb{N}, \mathbb{C})$. Consider

$F= \{x=(x_k)_{k\in \mathbb{N}}\in l^2(\mathbb{N}, \mathbb{C})|x_{2k}=x_{2k+1}\} $

Decompose $x$ into $F$ and $F^{\perp }$.

I proved $F$ is a closed subspace of $l^2(\mathbb{N}, \mathbb{C})$,

$F^{\perp }= \{y \in l^2(\mathbb{N}, \mathbb{C}) | \sum_{k\in \mathbb{N}}x_{2k}(y_{2k}+y_{2k+1})= 0, x=(x_k)_k \in F \}$

and I think I need to use the projection theorem in Hilbert space but I couldn't decompose $x$. Could you help me, please?

Answer 1

Write any $(x_n)$ as the sum of $(x_1,\frac {x_2+x_3} 2, \frac {x_2+x_3} 2,\frac {x_4+x_5} 2,\frac {x_4+x_5} 2,\cdots )$ and $(0,\frac {x_2-x_3} 2, \frac {x_3-x_2} 2,\frac {x_4-x_5} 2,\frac {x_5-x_4} 2,\cdots )$. Verify that the first sequence is in $F$ and that the second one is orthogonal to $F$. This decomposition is the one we want.