The Space of Polynomials is Incomplete with Respect to the Summed Coeffcient Norm I'm working on the following question in Royden:

Let $X$ be the linear space of all polynomials defined on $\mathbb{R}$. For $p \in X$, define $\|p\|$ to be the sums of the absolute values of the coefficients of $p$. Show that this is a norm on $X$. For each $n$ define $\psi_n: X \to \mathbb{R}$ by $\psi_n(p)=p^{(n)}(0)$. Use the properties of the sequence $\psi_n$ in $L(X,\mathbb{R})$ to show that $X$ is not a Banach Space.

Showing the norm part is straight forward. I want to say something like -- there is a pointwise bound but no uniform bound, so by Banach-Steinhaus, $X$ is not complete. Here's my half-solution:
We can establish pointwise bounds like so:
$$\psi_n(p) \leq c_0 \text{ (the constant coeffcient in the polynomial)} \leq \|p\|$$
To get a failure of uniform bound I want to choose a particularly bad $p \in X$ so that there's an $n$ in the LHS of the last inequality:
$$|\psi_n(p)| \leq \|p\| \|\psi_n\| \implies \frac{|\psi_n(p)|}{\|p\|} \leq  \|\psi_n\| \implies \frac{|p^{(n)}(0)|}{\|p\|} \leq  \|\psi_n\|$$
Do people see what might work? Or do I need a different argument?
There a similar question here, but it didn't help me.

For any $p$, the sequence $\psi_n(p)$ is constantly equal to $0$ as soon as $n$ is greater than the degree of $p$. In particular, $\sup_n(|\psi_n(p)|) < \infty$.
On the other hand, $||\psi_n|| \geq n!$ since $\frac{|\psi_n(x^n)|}{|x^n|} = n!$.
It follows that $X$ does not satisfy the conclusion of the Banach-Steinhaus theorem and thus cannot be a Banach space.
