Riemann Integral Operator is Continuous Linear Transformation
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Theorem
Let $\struct {C \closedint a b, \norm {\, \cdot \,}_\infty}$ be the normed vector space of real-valued functions continuous on $\closedint a b \subseteq \R$ equipped with the supremum norm.
Let $T : C \closedint a b \to \R$ be the Riemann integral operator:
- $\ds \forall \mathbf x \in C \closedint a b : \map T {\mathbf x} = \int_a^b \map {\mathbf x} t \rd t$
Then $T$ is a continuous mapping.
Proof
We have that Integral Operator is Linear.
Furthermore:
\(\ds \size {T \mathbf x}\) | \(=\) | \(\ds \size {\int_a^b \map {\mathbf x} t \rd t}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \int_a^b \size {\map {\mathbf x} t} \rd t\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \int_a^b \norm {\map {\mathbf x} t}_\infty \rd t\) | Definition:Supremum Norm on Space of Continuous on Closed Interval Real-Valued Functions | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {b - a} \norm {\mathbf x}_\infty\) |
By Continuity of Linear Transformation between Normed Vector Spaces, $T$ is continuous.
$\blacksquare$
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.3$: The normed space $\map {CL} {X,Y}$