19.3.32 Linear operators

Condition: Find the norm of the operator \( A: X \rightarrow X \), \( A p(t)=p^{\prime}(t)+\int_{0}^{t} p(s) d s \), if: a) \( X=\left(P^{1}[0,1],\|\cdot\|_{2}\right), \quad\|x\|_{2}=\sqrt{\int_{0}^{1} x^{2}(t) d t} \), b) \( X=\left(P^{1}[0,1],\|\cdot\|_{\infty}\right), \quad\|x\|_{\infty}=\max _{0 \leq t \leq 1}|x(t)| \). Hint: Solve the corresponding optimization problem: \[ \|A\|=\sup _{\|x\| \leq 1}\|A x\|, \quad x(t)=a t+b \in P^{1}[0,1] \text {. } \]