Projection in $\ell^2 (\mathbb{N})$ Let $X=\ell^2(\mathbb{N})$ and $C=\left\{x=(x_k) \in X: x_0+x_1+x_2=1\right\}$. I have prooved that $C$ is a non-empty, convex and closed subsect of $X$. Now I want to find the projection of the element $0=(0,0,\dots) \in \ell^2(\mathbb{N})$. Can you help me?

Since the projection in Hilbert space gives the nearest point, you are looking to minimize the quantity $x_0^2+x_1^2+x_2^2$ subject to the condition $x_0+x_1+x_2=1$, and it is not too hard to prove that the minimum occurs when $x_0=x_1=x_2=1/3$.
