The following is a homework question and I'd be glad if you could tell me if I did it right:
Question:
Consider the space of real polynomials $\mathbb{R}[X]$. Define the family of norms || p ||_s := || p ||_{C([0,s])} = \sup_{x \in [0,s]} |p(x)| for any $s \in \mathbb{R}_{>0}$. Use the Stone-Weierstrass theorem to show that these are all inequivalent.
Answer: They are not actually all equivalent. There are two cases:
(i) $s \in (0,1)$:
Then $\sup_{x \in [0,s]} |p(x)| \in \mathbb{R}$ so $\exists K : |p(x)| \leq K$.
So for $s, s^\prime$, there exist $K, K^\prime $ such that || p ||_s K \leq || p ||_{s^\prime} \leq || p ||_s K^\prime
(ii) $s \geq 1$:
Consider $p(x) = x^n$. Then $ \lim_{n \rightarrow \infty} || x^n ||_s = \lim_{n \rightarrow \infty} s^n = \infty$ So there are no $K,K^\prime$ such that the above holds.
I didn't use Stone-Weierstrass, so the answer is probably wrong but I don't see where. Many thanks for your help!