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

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