orthonormal basis in $l^{2}$ I need an orthonormal basis in $l^{2}$. One possible choice would be to take as such the sequences $\{1,0,0,0,...\}, \{0,1,0,0,...\}, \{0,0,1,0,...\}$, but I need a basis where only finitely many components of the basis vectors are zero. Does anyone know a way to construct such a basis? One possible vector for such a basis would be $\{1,1/2,1/3,1/4,...\}$ devided by its norm. However, I don't know how to find similar vectors that are orthogonal to this one and to each other.

How about this: consider $w = (1,1/2,1/3,1/4,\dots)$ (or any other vector with no no-zero entry), and complement it with all the standard basis vectors $e_1$, $e_2$,
$\dots$ to a complete set and apply Gram-Schmidt orthonormalization?
