Let $P_M =$ {polynomials of degree at most $M+1$}. Then $P_M$ has basis $\{1, x, x^2, \ldots , x^M\}$ and is a finite-dimensional vector space over $\Bbb R$.
My question is as follows: I want to show that $P_M$ is complete. Now, $P_M$ is a subspace of the infinite dimensional vector space $C(\Bbb R)$, the set of all continuous function $\Bbb R \rightarrow \Bbb R$. To show that $P_M$ is complete, I have to equip $C(\Bbb R)$ with a norm, but what norm can I equip? Since $C(\Bbb R)$ contains unbounded functions, the $sup$ norm won't work (as far as I understand).
Is it possible to restrict $P_M$ to be a field over $[0,1]$, to which I can then equip the $sup$ norm? We then have that $P_M$ is a subspace of $C[0,1]$, and thus complete. Will my result then hold for $P_M$ over $\Bbb R$?
Hints only please, if possible!
If you just want to make $P_M$ a complete space you can use any norm on it because all finite dimensional spaces are Banach spaces. If you want to consider it as a closed subspace of $C(\mathbb R)$ you can make the latter a complete metric space. There is no norm you can think of so define $d(f,g) =\sum_{n=1}^{\infty } \frac {||f-g||_n} {2^{n}(1+||f-g||_n)}$ where $||f-g||_n=\sup \{|f(x)-g(x):0 \leq x \leq n\}$ With some effort you can show that this makes $C(\mathbb R)$ a complete metric space and $P_M$ is a closed subspace.