Could you help me with the following?
We are given a linear map:
$$ \phi: \ \mathcal{P} \ni p \rightarrow q \in \mathcal{P}, \ \ \ q'=p, \ \ \ q(0) = 0$$
Here $\mathcal{P}$ is the space of real polynomials.
In $\mathcal{P}$ we consider four different norms:
$$3) ||p||^1_{\infty} = |p(0)| + \sup _{t \in [0,1]}|p'(t)|$$ $$4) ||p||_1^1 = |p(0)| + \int_0^1|p'(t)|dt$$
I've already solved norms 1),2).
Here is how I approach this:
$$3) ||\phi p||_{\infty}^1 = |q(0)| +\sup _{t \in [0,1]}|q'(t)| = \sup _{t \in [0,1]}|p(t)| \le |p(0)| + \sup _{t \in [0,1]}|p(t)| \le \sup _{t \in [0,1]}|p'(t)| \le ||p||^1_{\infty}$$
So let's take $p(t) = t$, then $q(t) = \frac{1}{2}t^2$ and so: $||p||_{\infty}^1 = 0+1=1$, $||\phi p||_{\infty}^1 =||q||_{\infty}^1 = 0+1=1$.
So the norm 3) seems to be done.
So it seems we can estimate it better.
But I don't see how. I have the same problem with norm 4).
Could you help me with this?
$\|\phi p\|_1^1=|q(0)|+\int_0^1|q'(t)|dt=\int_0^1|p(t)|dt.$
Now we have $|p(t)|=|p(0)+\int_0^tp'(u)du|\leq |p(0)|+\int_0^t|p'(u)|du\leq |p(0)|+\int_0^1|p'(u)|du=\|p\|_1^1.$
So $\|\phi p\|_1^1=\int_0^1|p(t)|dt\leq \int_0^1\|p\|_1^1dt=\|p\|^1_1.$ Therefore $\|\phi\|^1_1\leq 1.$
If $p$ is a non-zero constant then $\|\phi p\|^1_1=\|p\|^1_1=|p(0)|\ne 0.$ Therefore $\|\phi \|^1_1\geq 1.$