Let $\mathbb{C}[x_1,\ldots,x_n]$, $n \geq 1$.
Is it possible to find a norm on $\mathbb{C}[x_1,\ldots,x_n]$? (If not, why?).
Where a norm $\rho$ is a function $\rho: \mathbb{C}[x_1,\ldots,x_n] \to \mathbb{R}$, having the following properties: $\forall c \in \mathbb{C}, u,v \in \mathbb{C}[x_1,\ldots,x_n]$,
(1) $\rho(v) \geq 0$. $\rho(v)=0$ if and only if $v=0$
(2) $\rho(cv)= |c|\rho(v)$
(3) $\rho(u+v) \leq \rho(u)+\rho(v)$
Any comments are welcome!
I have a problem with the second property, which ruins some ideas I had in mind.
The polynomial structure is a red herring; you only care about its vector space structure, and $\mathbb{C}[x_1, \ldots, x_n]$ is a complex vector space of countable dimension.
Another description of this space is that it is the vector space of all complex sequences that have only finitely many nonzero terms.
As an example, any of the $L^p$ norms would work: for real $p \geq 1$
$$ \| \vec{a} \|_p = \left( \sum_{n} | a_n |^p \right)^{1/p} $$
defines a norm, as does,
$$ \| \vec{a} \|_{\infty} = \max_n\left( |a_n| \right) $$
Basically the same description works to define these norms on polynomials directly, rather than the alternate representation in terms of sequences: just take the sums over the coefficients of the polynomial rather than terms of a sequence.