Riemann Integral Operator is Linear Mapping

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

$\blacksquare$

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.1$: Continuous and linear maps. Linear transformations