Riemann Integral Operator is Linear Mapping

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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

$\blacksquare$

Sources