How to Prove Linearity and Compute the Norm of $T_K$ for a Compact Kernel in $\mathbb{R}^d$?

Question

Hi all this is my question: Let $U, V \subset \mathbb{R}^d$ be compact, $K \in C(U \times V)$ and $T_K: C(V) \rightarrow C(U)$ given by $$ T_K(u)(x)=\int_V K(x, y) u(y) d y . $$ (i) Show that $T_K \in \mathrm{L}(C(V), C(U))$ and compute $\left\|T_K\right\|$.

And, this is my solution:

Showing $T_K \in \mathrm{L}(C(V), C(U))$ :

1. Linearity:

• To show that $T_K$ is linear, we must prove that for any $u_1, u_2 \in C(V)$ and scalars $\alpha, \beta$, the following holds: $$ T_K\left(\alpha u_1+\beta u_2\right)(x)=\alpha T_K\left(u_1\right)(x)+\beta T_K\left(u_2\right)(x) $$
• Computing both sides yields: $$ \begin{aligned} & T_K\left(\alpha u_1+\beta u_2\right)(x)=\int_V K(x, y)\left(\alpha u_1(y)+\beta u_2(y)\right) d y \\ & =\alpha \int_V K(x, y) u_1(y) d y+\beta \int_V K(x, y) u_2(y) d y \\ & =\alpha T_K\left(u_1\right)(x)+\beta T_K\left(u_2\right)(x) \end{aligned} $$
• Hence, $T_K$ is linear.

2. Continuity:

• To show that $T_K$ is continuous, we need to show that it is bounded, i.e., there exists a constant $C$ such that $\left\|T_K(u)\right\|_{C(U)} \leq C\|u\|_{C(V)}$ for all $u \in C(V)$.
• Since $K$ is continuous on the compact set $U \times V, K$ is bounded. Let $M=$ $\sup \{|K(x, y)|:(x, y) \in U \times V\}$.
• For $u \in C(V)$ and $x \in U$, we have: $$ \begin{aligned} & \left|T_K(u)(x)\right|=\left|\int_V K(x, y) u(y) d y\right| \leq \int_V|K(x, y) u(y)| d y \\ & \leq \int_V M|u(y)| d y \leq M\|u\|_{C(V)} \mu(V) \end{aligned} $$ where $\mu(V)$ denotes the measure of $V$.
• Therefore, $\left\|T_K(u)\right\|_{C(U)} \leq M \mu(V)\|u\|_{C(V)}$, and $T_K$ is bounded.

Computing $\left\|T_K\right\|$ : The operator norm of $T_K$ is defined as: $$ \left\|T_K\right\|=\sup \left\{\left\|T_K(u)\right\|_{C(U)}: u \in C(V),\|u\|_{C(V)} \leq 1\right\} $$

From the continuity proof, we have: $$ \left\|T_K(u)\right\|_{C(U)} \leq M \mu(V)\|u\|_{C(V)} $$

Hence, the operator norm $\left\|T_K\right\|$ is at most $M \mu(V)$. Since this is the least upper bound for the norm of $T_K(u)$ for all $u$ with norm less than or equal to 1, we conclude that: $$ \left\|T_K\right\|=M \mu(V) $$

Thus we have shown that $T_K$ is a linear and continuous operator from $C(V)$ to $C(U)$ and computed its norm as $M \mu(V)$, where $M$ is the bound on the kernel function $K$ and $\mu(V)$ is the measure of the set $V$.

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I don't have a specific question but I am not sure if everything I did is right. I would greatly appreciate if you can proofread it. Thanks :)