continuity of functional in topology given by seminorms

Question

In $\mathbb{R}[x]$ we consider a family of seminorms:

$$\rho_n(f)=\frac{1}{n}\sum_{k=1}^{n}|f(\frac{k}{n})|, \ \ \ n\in\mathbb{N}$$

I checked that this family is separated. No I don't know how to show that the functional $$\varphi: \mathbb{R}[x]\rightarrow \mathbb{R}, \ \varphi(f)=\int_0^1 f(x)dx, \ \ f\in\mathbb{R}[x]$$ is contiunous in topology given by above family.

Answer 1

It isn't continuous: As $\varphi$ is linear, continuity (at $0$) means that there are $n\in \mathbb N$ and $c>0$ such that $|\varphi(f)|\le c\rho_n(f)$. This can be shown to be wrong by choosing an interpolation polynomial $f$ with $f(k/n)=0$ for $k\in\{0,\ldots,n\}$ and $f(1/(n+1))=1$.