Which of the following defines norm on V?

Question

Let $V$ denote the vector space of all polynomials over $\mathbb R$ of degree less than or equal to $n$. Which of the following defines a norm on $V$.

1. $\|p \|^2 = |p(1)|^2 + \cdots + |p(n+1)|^2, p \in V $.

2. $\| p \| = \sup_{t \in [0,1]} |p(t) |, \ \ p\in V$.

3. $\| P \| = \int_0^1 |p(t)|\, dt , \ \ p \in V$

4. $\| p \| = \sup_{t \in [0,1]} |p'(t) |, \ \ p\in V$.

I think (1), (2), (3) and (4) satisfy all the properties of norm. I have done this problem, I want only to confirm that my answer is right or not.

Answer 1

The answer is not right. (I do not elaborate, because you only asked for confirmation.)

Answer 2

1. $\|p\|=0$ implies $p(1)=p(2)=\cdots =p(n+1)=0$. That is the polynomial $p$ has $n+1$ zeros ; which is possible only when $p \equiv 0$ , because $p$ is of degree at most $n$. Also , if $p\equiv 0$ then clearly $\|p\|=0$.

$\|\alpha p\|=|\alpha|.\|p\|$ ,very clear ! Triangle inequality follows from Minkowski's inequality . Hence , it defines a norm.

2. is $\sup$ norm and 3. is integral-norm , they are standard norms !

4. If you take $p$ as any non-zero constant then $\|p\|=0$. So it does not satisfy the property of norm.