Operator of Integrated Weighted Derivatives is Linear Mapping

Theorem
Let $n \in \N$.

Let $I := \closedint a b$ be a closed real interval.

Let $\map {a_i} x : I \to \R$ be Riemann integrable functions.

Let $f, g \in \map {C^n} I$ be Riemann integrable real-valued functions of differentiability class $k$.

Let $L : \map {C^n} I \to \R$ be the operator of integrated weighted derivatives.

Then $L$ is a linear mapping.