In a paper I was reading, it was mentioned that if $M$ is a closed Riemannian manifold, then by fixing a basis for $L^2(M)$, the space of smoothing operators on $L^2(M)$ can be identified with the algebra of matrices $a_{ij}$ such that
$$\sup_{i,j}i^k j^l |a_{ij}| <\infty$$
for all $k,l\in\mathbb N$.
Question 1: Could someone elaborate on how this identification can be done, or point me to a reference?
Here's a simpler question. Let $\Delta$ be the Laplacian on $M$, and let $\{u_i\}_{i\in\mathbb{N}}$ be an orthonormal basis of eigenvectors of $\Delta$. Then we can identify $L^2(M)$ with $l^2(\mathbb{N})$ using this basis, and let us denote this isomorphism by $$\phi\colon l^2(\mathbb{N})\to L^2(M).$$
Let $B$ denote the space of sequences $b_i$ such that
$$\sup_i i^k|b_i|<\infty$$
for all $k\in\mathbb{N}$.
Question 2: Is the space $C^\infty(M)$ of smooth functions on $M$ equal to $\phi(B)$?