# Riemann Integral Operator is Linear Mapping

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## Theorem

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

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

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

Then $I$ is a linear mapping.

## Proof

Let $x, y \in C \closedint a b$ 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 $\ds$ $=$ $\ds \int_a^b \map x t \rd t + \int_a^b \map y t \rd t$ Linear Combination of Definite Integrals $\ds$ $=$ $\ds \map I x + \map I y$ 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 $\ds$ $=$ $\ds \alpha \int_a^b \map x t \rd t$ Primitive of Constant Multiple of Function $\ds$ $=$ $\ds \alpha \map I x$ Definition of Riemann Integral Operator

$\blacksquare$