# 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.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.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 transformations:

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

$\blacksquare$

## Sources

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