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