Find the norms of the linear operator $Af(x) =f(x^2)$

Question

Find the 1-norms of the linear operator $Af(x) =f(x^2)$ if:

a) $A: C[0,1] \rightarrow C[0,1]$

b) $A: C[-1,1] \rightarrow C[0,1]$

c) $A: L^1(0,1) \rightarrow L^2(0,1)$

I honestly have no idea how to do this. For part a, my guess is: $\|Af\|= \|f(x^2)\| \leq \|f(x)\| \forall f \in C[0,1] $? then I don't know what to do from here.

For part b, I think you get the same inequality except here its $\forall f \in C[-1,1]$, but again I don't know how to conclude this.

For part c, we have $\|Af(x)\|_1 \leq \|f(x^2)\|_1$

I keep seeing examples online where they just let be equal to some value such as $f \equiv 1 $ in order to show $\|A\| \geq$ whatever bound we get from above, but I don't understand why that's ok or if I can do the same here.

EDIT: I understand parts a and b, but I'm still confused on part c

Answer 1

For a) and b), the answer is $1$. It is clear that $\|Af\|\leqslant\|f\|$ and, if you take $f=1$, you get that $\|Af\|=\|f\|=1$.