Space of real polynomials of one variable isn't complete

Question

Consider $E$ - the vector space of all real polynomials of one variable.

I need to prove that it is not complete under any norm.

I was thinking I could use the fact that certain functions, for example $ \exp x$ can be approximated by a sequence of polynomials $P(x) = (1 + \frac{x}{n})^n, \ \ n \in \mathbb{N}$.

This approach doesn't seem to depend of any particular norm, does it?

Could you tell me if I'm right or if I'm missing something?

Thanks a lot!

Answer 1

i'm a bit out of my depth here, so apologies if this is nonsense, but i think your idea works. suppose $\|x\|=N$ and define $f_n=\sum_{j=0}^n \frac{x^j}{j!}$. this sequence converges absolutely for all $N$, but is not a polynomial.