In Chapter 1 Section 1 of Modern Fourier Analysis by Grafakos, we consider the space of tempered distributions modulo polynomials, $\mathscr{S}'(\mathbb R^n)/\mathscr{P}(\mathbb R^n)$, which coincides with the space $\mathscr{S}_0'(\mathbb R^n)$, where $\mathscr{S}_0(\mathbb R^n)$ is the set of $\varphi\in\mathscr{S}(\mathbb R^n)$ such that $$\int_{\mathbb R^n}\!x^\alpha\varphi(x)\,\mathrm dx=0$$ for all multiindices $\alpha$.
However, I was wondering what can be said about the induced topology on $\mathscr{P}(\mathbb R^n)\subset\mathscr{S}'(\mathbb R^n)$. It is clear that any sequence of polynomials $p_N\in\mathscr{P}(\mathbb R^n)$ which is bounded in degree and whose coefficients all converge, will converge in $\mathscr{S}'(\mathbb R^n)$. But is the converse true? Does convergence of polynomials in $\mathscr{S}'(\mathbb R^n)$ imply boundedness in degree? Clearly this is not true in $\mathcal{D}'(\mathbb R^n)$, so if this were the case, it would be a consequence of the fact that Schwartz functions need not have compact support.
Maybe this would be best proved in Fourier space?
Yes.
Say $$\rho_{n,N}(f)=\sup_{|\alpha|\le N}\sup_{x\in\mathbb R^d}(1+|x|)^n|D^\alpha f(x)|,$$so $\rho_{n,N}$ is one of the seminorms defining the topology on the Schwarz space. It follows from the Uniform Boundedness Principle (a Frechet space version) that if $u_k\to u$ in $\mathscr S'$ there exist $N,n,c$ so $$|u_k(\phi)|\le c\rho_{n,N}(\phi).$$