The space $(\mathbb{Q}_{p}[X],|-|_{\text{gauss}})$ is not complete

Question

I'm stuck trying to prove incompletness of certan $p$-adic space. Let $\mathbb{Q}_{p}[X]$ the space of polynomials. We know that has infinite dimension. On $\mathbb{Q}_{p}[X]$ define the function $\sum a_{n}X^{n}\mapsto\sup_{i}|a_{i}|_{p}$, called the Gauss norm, and denoted by $|-|_{\text{gauss}}$.

Before the main theorem of the section (which states basically that all nonarchimedean norms are equivalent if the normed space is of finite dimension, and, consequently the normed space is complete) of certain book about $G$-function, the author says that $(\mathbb{Q}_{p}[X],|-|_{\text{gauss}})$ is a non-complete normed space (so, the theorem is true always in the finite dimensional case). It is not clear to me this fact, so I tried to see why. Here is my attempt:

Consider the sequence of polynomials: $s_{n}(X)=1+pX+p^{2}X^{2}+\cdots+p^{n}X^{n}$. Since $|s_{n+1}-s_{n}|_{\text{gauss}}=p^{-n-1}\rightarrow 0$, we see that is Cauchy. So, suppose, in order to get a contradiction, that there is a limit in $\mathbb{Q}_{p}[X]$, say $Q(X)=a_{0}+a_{1}X+\cdots+a_{m}X^{m}$.

I'm tempted to the following: $$0=\lim|s_{n}-Q|_{\text{gauss}}=|\lim(s_{n}-Q)|_{\text{gauss}}=|\frac{1}{1-pX}-Q|_{\text{gauss}},$$ so $Q=1/(1-pX)\in\mathbb{Q}[X]$ which cannot be possible. Now, I'm worried about if I can go from a $p$-adic limit to a $X$-adic limit.

Is this approach correct?

Also, following the definition of convergence: for all $\varepsilon>0$, there is an $N\in\mathbb{N}$, such that $|s_{n}-Q|_{\text{gauss}}<\varepsilon$ for every $n\geq N$. In other words, $\sup_{n\geq N}|p^{n}-a_{n}|<\varepsilon$. If $N\leq m$ I can say that some of the $a_{i}$ are powers of $p$, but I can't go further. If $N>m$, I can't say anything about the $a_{i}$, only that $a_{i}=0$.

Can anyone give me a hint? I would appreciate it

Thank you

Answer 1

To turn my comment into an answer, assume that there exists $Q = \displaystyle \lim_{n\to \infty} s_n$ in $\mathbb Q_p[X]$, and let $m := \deg Q$. Then for $n \ge m+1$, we have $\lvert s_n -Q\rvert_{gauss} \ge \lvert p^{m+1}-0\rvert_p = p^{-(m+1)}$ which contradicts $\lim_{n\to \infty} \lvert s_n -Q\rvert_{gauss} = 0$. Since as you write the sequence $(s_n)_n$ is Cauchy, we thus have that the space is not complete with respect to the Gauß norm.

Your attempt is incomplete because you have to define what "$\dfrac{1}{pX-1}$" is supposed to mean. Is it "the unique element in the ring which is the inverse of $pX-1$"? Well, such an element does not exist in $\mathbb{Q}_p[X]$; it does exist in various rings which contain $\mathbb Q_p[X]$, among them the power series ring $\mathbb Q_p[[X]]$, but also e.g. in $\mathbb Q_p(X)$. One can of course show that $\mathbb Q_p[[X]]$ is (canonically isomorphic to) the completion of $\mathbb Q_p(X)$ with respect to the Gauß norm, and then it's clear that the limit of $s_n$ is not a polynomial (and one does not even need that it is the inverse of something). But to show this feels like the step to do after realising that the space $\mathbb Q_p[X]$ is not complete.

Finally, if one worked with the ring $\mathbb Q_p(X)$ instead, it would be awkward to even try to extend the Gauß norm to that ring.

Answer 2

Define $A_n$ to be the polynomials in $\mathbb{Q}_p[X]$ of degree $\leq n$. Then $A_n$ is a closed subspace of $\mathbb{Q}_p[X]$ and we have $$ \mathbb{Q}_p[X] =\bigcup_{n\geq 0} A_n.$$ If we assume that $\mathbb{Q}_p[X]$ is complete with respect to the Gauss norm, then we can apply the Baire category theorem and get that some $A_n$ has nonempty interior. However, nonempty interior means for a subspace that it is the whole space which gives you the desired contradiction.