I was reading a note on distributions. The author left the following Schwartz representation theorem as an exercise:
I'm trying to prove the theorem. According to the hints, I've done all the procedure except the last line: how could we get $$u=\sum_{|\alpha|+|\beta|\leq 2n+2+k} x^\beta D^\alpha u_{\alpha,\beta}$$ once we know $R_{n+1}(R_{n+1+k}^tu)$ is bounded and continuous?
ps: there is a typo in the picture, $R_p^tu(\phi)$ not $T_p^tu(\phi)$