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) \ (a) \ (a) \ (a) \ ( X = \ left (p^{1} [0.1], \ | \ cdot \ | _ {2} \ right), \ quad \ | x \ | _ {2} = \ sqrt {\ int_ {0} {1} x^{2} (t) d t} \), b), b), b), b), b), b), b), b) b) \ (X = \ left (p^{1} [0.1], \ | \ cdot \ | _ {\ infty} \ right), \ Quad \ | _ {\ Infty} = \ MAX _ {0 \ LeQ T \ LeQ 1} | X (T) | \). Indication: Solve the corresponding optimization problem: \ [\ | a \ | = \ SUP _ {\ | x \ | \ leq 1} \ | a x \ |, \ quad X (t) = a t+b \ in p^{1} [0.1] \ text {. } \]