Convergence of a series in the sense of distributions

Question

Let $(e_i)_{i \geq 0}$ an Hilbertian basis of $L^2(\mathbb{R})$

Let $(a_i)_{i \geq 0}$ a real sequence and $T$ the distribution over $\mathbb{R}$ defined by : $\langle T, \varphi \rangle = \sum_{i = 0}^{\infty} a_i \langle e_i, \varphi \rangle $

We suppose $T = 0$. Is it true that $a_i = 0$ for every $i$?

I tried to take $\varphi$ close to $e_0$ but I don't manage to conclude...

Thank you in advance !

Answer 1

If $T=0$, then $\langle T, \varphi \rangle = 0$ for all $\varphi \in L^2(\mathbb R)$. This means $\langle T, e_j \rangle = 0$ for all $j \geq 0$.
Can you go on from there?