# Riemann Integral Operator is Linear Mapping

## Theorem

Let $\CC \closedint a b$ be the space of continuous Riemann integrable functions.

Let $\R$ be the set of real numbers.

Let $I : \CC \closedint a b \to \R$ be the Riemann integral operator.

Then $I$ is a linear mapping.

## Proof

Let $x,y \in \CC \closedint a b$ be Riemann integrable.

Let $\alpha \in \R$.

### Distributivity

 $\ds \map I {x + y}$ $=$ $\ds \int_a^b \paren{\map x t + \map y t} \rd t$ Definition of Riemann Integral Operator $\ds$ $=$ $\ds \int_a^b \map x t \rd t + \int_a^b \map y t \rd t$ Sum Rule $\ds$ $=$ $\ds \map I x + \map I y$ Definition of Riemann Integral Operator

$\Box$

### Positive homogenity

 $\ds \map I {\alpha x}$ $=$ $\ds \int_a^b \alpha \map x t \rd t$ Definition of Riemann Integral Operator $\ds$ $=$ $\ds \alpha \int_a^b \map x t \rd t$ Multiple Rule $\ds$ $=$ $\ds \alpha \map I x$ Definition of Riemann Integral Operator

$\blacksquare$