Finding the orthogonal projection on a subspace $K \in \ell^2(\mathbb{N})$

Question

I'm trying to find the orthogonal projection of $\ell^2(\mathbb{N})$ on $K$ and then determining $p_K(x)(2n)$ for $x\in \ell^2(\mathbb{N})$ and $n\in \mathbb{N}$. The subspace K is defined as $$ K = \{x\in \ell^2(\mathbb{N})|x(2n+1) = x(2n) \text{ for all } n \in \mathbb{N} \} $$ and $\ell^2(\mathbb{N}) = \{ x: \mathbb{N} \rightarrow \mathbb{C} \mid\sum_{n=0}^\infty |x(n)|^2 < \infty \}$.

I'm sure the problem is much simpler than what I think, I am just completely lost in this topic at the moment. Can anyone help?

Answer 1

Hint: Consider $e_0=\left(\frac1{\sqrt2}, \frac1{\sqrt2}, 0,0,\dots\right),\ e_1=\left(0,0,\frac1{\sqrt2},\frac1{\sqrt2},0,0,\dots\right),\,\dots$, prove that it's an orthonormal basis of $K$, and then verify that $$p_K(x)=\sum_n\langle x,e_n\rangle e_n\,.$$

You should receive that $p_K(x)(2n)=\dfrac{x_{2n}+x_{2n+1}}2$.