19.3.4 Linear operators

Problem: Prove the linearity and calculate the norm of the functional \( f(x)=\int_{0}^{2} t^{2} x(t) d t \), where a) \( x \in L_{1}[0,2] \), b) \( x \in L_{2}[0,2] \), c) \( x \in C[0,2] \).