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

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, $$S(x)=(0,0,a_1,0,a_2,0,a_3,0,a_5,...)$$

• Yes, your solution is correct. – Kavi Rama Murthy May 13 at 5:43

$$l^2(N)$$ 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.

• $l^2(N)$ is the space all summable sequences of the form $\sum_{k=1}^\infty a_k < \infty$, with the inner product $<x,y>=\sum_{k=1}^\infty x_k y_k$. Does it answer your confusion? – A Slow Learner May 13 at 5:25
• OK then. You are correct. – Hume2 May 13 at 5:32