# 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.

## Proof

### Distributivity

 $\ds \map L {f + g}$ $=$ $\ds \int_a^b \sum_{i \mathop = 0}^n \map {a_i} x \dfrac {\d^i}{ {\d x}^i} \paren{\map f x + \map g x} \rd x$ Definition of Operator of Integrated Weighted Derivatives $\ds$ $=$ $\ds \int_a^b \sum_{i \mathop = 0}^n \map {a_i} x \paren{\map {f^{\paren i} } x + \map {g^{\paren i} } x} \rd x$ Linear Combination of Derivatives $\ds$ $=$ $\ds \int_a^b \sum_{i \mathop = 0}^n \map {a_i} x \map {f^{\paren i} } x \rd x + \int_a^b \sum_{i \mathop = 0}^n \map {a_i} x \map {g^{\paren i} } x \rd x$ Linear Combination of Definite Integrals $\ds$ $=$ $\ds \map L f + \map L g$ Definition of Operator of Integrated Weighted Derivatives

$\Box$

### Positive homogenity

 $\ds \map L {\alpha f}$ $=$ $\ds \int_a^b \sum_{i \mathop = 0}^n \map {a_i} x \dfrac {\d^i}{ {\d x}^i} \paren {\alpha \map f x} \rd x$ Definition of Riemann Integral Operator $\ds$ $=$ $\ds \int_a^b \sum_{i \mathop = 0}^n \map {a_i} x \alpha \map {f^{\paren i} } x \rd x$ Linear Combination of Derivatives $\ds$ $=$ $\ds \alpha \int_a^b \sum_{i \mathop = 0}^n \map {a_i} x \map {f^{\paren i} } x \rd x$ Primitive of Constant Multiple of Function $\ds$ $=$ $\ds \alpha \map L f$ Definition of Riemann Integral Operator

$\blacksquare$