General Formula of a Linear Operator given its act on the Standard Orthonormal Basis

Question

I am trying to find a general formula for a linear operator on a Hilbert space when its action on the standard orthonormal basis is known.

I include my work below. Please tell me whether my solution is correct.

Let $S(e_k)=e_{2k+1}$ be a linear operator in the Hilbert space $l^2(N)$, where, $\{e_k\},k=0,1,2...$, is the standard orthonormal basis.

To find a formula for $S(x)$ where $x=(a_1,a_2,a_3,...)$, below is my work: $$ S(x)=S(\sum_{k=1}^\infty a_ke_k)=\sum_{k=1}^\infty a_kS(e_k)=\sum_{k=1}^\infty a_ke_{2k+1}=a_1e_3+a_2e_5+a_3e_7+... $$ So, $$ S(x)=(0,0,a_1,0,a_2,0,a_3,0,a_5,...) $$

Answer 1

$l^2(N)$ is not a Hilbert space. I think, you meant $L^2(N)$.

I don't know what you mean by standart orthonormal basis in $L^2(N)$. Do you mean the Fourier basis? It doesn't matter actually, it works with any orthogonal basis.

So yes, I think, you are correct except these tiny details.

EDIT: You are correct actually.