# Linear Integral Bounded Operator is Continuous

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## Theorem

Let $I = \closedint 0 1$ be a closed real interval.

Let $A : I \times I \to \R$ be a real function such that:

- $\ds \int_0^1 \int_0^1 \paren {\map A {t, \tau} }^2 \rd t \rd \tau < \infty$

Further research is required in order to fill out the details.In particular: For now "bounded" means above. Need to check if this meaning is standardYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by finding out more.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Research}}` from the code. |

where $\times$ denotes the cartesian product.

Let $T_A : \map {L^2} I \to \map {L^2} I$ be an integral operator such that:

- $\ds \map {\paren {T_A \mathbf x} } t := \int_0^1 \map A {t, \tau} \map {\mathbf x} \tau \rd \tau$

where $\mathbf x \in \map {L^2} I$, and $\map {L^2} I$ is the Lebesgue $2$-space.

Then $T_A$ is a continuous transformation.

## Proof

We have that Riemann Integral Operator is Linear Mapping.

Further research is required in order to fill out the details.In particular: Probably this should be replaced with Lebesgue integral. The source does not say anything about compatibility of Riemann integral and Lebesgue spaceYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by finding out more.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Research}}` from the code. |

Hence, $T_A$ is a linear transformation.

Furthermore:

\(\ds \norm {T_A \mathbf x}_2^2\) | \(=\) | \(\ds \int_0^1 \paren {\int_0^1 \map A {t, \tau} \map {\mathbf x} \tau \rd \tau}^2 \rd t\) | Definition of P-Seminorm | |||||||||||

\(\ds \) | \(\le\) | \(\ds \int_0^1 \paren{\int_0^1 \paren{\map A {t, \tau} }^2 \rd \tau} \paren {\int_0^1 \paren{\map {\mathbf x} \tau}^2 \rd \tau }\rd t\) | Cauchy-Bunyakovsky-Schwarz Inequality for Definite Integrals | |||||||||||

\(\ds \) | \(\le\) | \(\ds \paren {\int_0^1 \int_0^1 \paren{\map A {t, \tau} }^2 \rd \tau \rd t} \paren {\int_0^1 \paren{\map {\mathbf x} \tau}^2 \rd \tau }\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \paren {\int_0^1 \int_0^1 \paren{\map A {t, \tau} }^2 \rd \tau \rd t} \norm {\mathbf x}_2^2\) | Definition of P-Seminorm | |||||||||||

\(\ds \) | \(<\) | \(\ds \infty\) |

Hence:

- $T_A \mathbf x \in \map {L^2} I$

By Continuity of Linear Transformation between Normed Vector Spaces:

- $T_A \in \map {CL} {\map {L^2} I}$

$\blacksquare$

## Sources

- 2017: Amol Sasane:
*A Friendly Approach to Functional Analysis*... (previous) ... (next): Chapter $\S 2.3$: The normed space $\map {CL} {X,Y}$