Integrated Linear Differential Mapping is Continuous

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:

where:


 * $\ds M : = \int_a^b \paren {\size {\map {\mathbf A} t} + \size {\map {\mathbf B} t} } \rd t$

By continuity of linear transformations $L$ is continuous.