Integrated Linear Differential Mapping is Continuous
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Theorem
Let $C^1 \closedint a b := \map {C^1} {\closedint a b, \R}$ be the space of real functions of differentiability class $C^1$.
Let $S$ be the set of differentiable functions on closed real interval vanishing at their endpoints:
- $S := \set {\mathbf h \in C^1 \closedint a b : \map {\mathbf h} a = \map {\mathbf h} b = 0}$
Let $S \subseteq C^1 \closedint a b$ be equiped with the $C^1$ norm.
Let $\mathbf A, \mathbf B \in C \closedint a b$ be continuous real functions.
Let $L : S \to \R$ be the integrated linear differential mapping:
- $\ds \map L {\mathbf h} = \int_a^b \paren {\map {\mathbf A} t \map {\mathbf h} t + \map {\mathbf B} t \map {\mathbf h'} t} \rd t$
where $\mathbf h \in S$.
Then $L$ is continuous.
Proof
We have that the Integrated Linear Differential Mapping is Linear.
For $\mathbf h \in S$ we have:
\(\ds \size {\map L {\mathbf h} }\) | \(=\) | \(\ds \size {\int_a^b \paren {\map {\mathbf A} t \map {\mathbf h} t + \map {\mathbf B} t \map {\mathbf h'} t }\rd t}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \int_a^b \size {\map {\mathbf A} t \map {\mathbf h} t + \map {\mathbf B} t \map {\mathbf h'} t } \rd t\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \int_a^b \paren {\size {\map {\mathbf A} t} \size {\map {\mathbf h} t} + \size {\map {\mathbf B} t} \size {\map {\mathbf h'} t} } \rd t\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \int_a^b \paren {\size {\map {\mathbf A} t} \norm {\mathbf h}_\infty + \size {\map {\mathbf B} t} \norm {\mathbf h'}_\infty } \rd t\) | Definition of Supremum Norm on Space of Continuous on Closed Interval Real-Valued Functions | |||||||||||
\(\ds \) | \(\le\) | \(\ds \int_a^b \paren {\size {\map {\mathbf A} t} \norm {\mathbf h}_{1, \infty} + \size {\map {\mathbf B} t} \norm {\mathbf h'}_\infty } \rd t\) | $\norm {\mathbf h}_{1, \infty} = \norm {\mathbf h}_\infty + \norm {\mathbf h'}_\infty \ge \norm {\mathbf h}_\infty$ | |||||||||||
\(\ds \) | \(\le\) | \(\ds \int_a^b \paren {\size {\map {\mathbf A} t} \norm {\mathbf h}_{1, \infty} + \size {\map {\mathbf B} t} \norm {\mathbf h'}_{1, \infty} } \rd t\) | $\norm {\mathbf h}_{1, \infty} = \norm {\mathbf h}_\infty + \norm {\mathbf h'}_\infty \ge \norm {\mathbf h'}_\infty$ | |||||||||||
\(\ds \) | \(\le\) | \(\ds \paren {\int_a^b \paren {\size {\map {\mathbf A} t} + \size {\map {\mathbf B} t} } \rd t} \norm {\mathbf h}_{1, \infty}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds M \norm {\mathbf h}_{1, \infty}\) |
where:
- $\ds M : = \int_a^b \paren {\size {\map {\mathbf A} t} + \size {\map {\mathbf B} t} } \rd t$
By Continuity of Linear Transformation between Normed Vector Spaces $L$ is continuous.
$\blacksquare$
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.3$: The normed space $\map {CL} {X,Y}$