Given $n \in \mathbb{N},$ let $V$ be the vector space of all polynomials $P:[0,1] \rightarrow \mathbb{R}$ of degree less than or equal to $n$
(i) Given $P \in V,$ let $\|P\|:=\max \left\{\left|a_{0}\right|, \ldots,\left|a_{n}\right|\right\},$ where $P(t)=a_{0}+$ $\cdots+a_{n} t^{n}, t \in[0,1] .$ Prove that $\|\cdot\|$ is a norm in $V$
(ii) Let $1 \leq p \leq \infty$ and $P \in V$ be such that $\|P\|_{L^{p}([0,1])} \leq 1$. Prove that there exists a constant $c=c(n)>0$ such that $\left|a_{k}\right| \leq c$ for all $k=1, \ldots, n .$ Hint: The vector space $V$ has finite dimension $n$
It is easy to prove i.But ii is really difficult for me,I have proved that V has finite dimension, But I don't know how to use it.Can someone give me a more detailed hint?Thanks in advance.
If $X$ is a finite-dimensional real or complex normed vector space then all norms on $X$ are uniformly equivalent: If $\|\cdot\|_a$ and $\|\cdot\|_b$ are norms for $X$ then there exist $c,d\in \Bbb R^+$ such that for all $x\in X$ $$c\|x\|_a\le\|x\|_b\le d\|x\|_a.$$