这是一份kcl伦敦大学学院 7CCMMS05作业代写的成功案
Many problems in classical mathematical physics can be handled by reformulating them in terms of integral equations. A famous example is the Dirichlet problem discussed at the end of this section. Consider the simple operator $K$, defined in $C[0,1]$ by
$$
(K \varphi)(x)=\int_{0}^{1} K(x, y) \varphi(y) d y
$$
where the function $K(x, y)$ is continuous on the square $0 \leq x, y \leq 1 . K(x, y)$ is called the kernel of the integral operator $K$. Since
$$
|(K \varphi)(x)| \leq\left(\sup {0 \leq x, y \leq 1}|K(x, y)|\right)\left(\sup {0 \leq y \leq 1}|\varphi(y)|\right)
$$
we see that
$$
|K \varphi|_{\infty} \leq\left(\sup {0 \leq x, y<1}|K(x, y)|\right)|\varphi|{\infty}
$$
7CCMMS05 COURSE NOTES ：
Let $\langle M, \mu\rangle$ be a measure space and $\mathscr{H}=L^{2}(M, d \mu$ ). Then $A \in \mathscr{L}(\mathscr{H})$ is Hilbert-Schmidt if and only if there is a function
$$
K \in L^{2}(M \times M, d \mu \otimes d \mu)
$$
$$
(A f)(x)=\int K(x, y) f(y) d \mu(y)
$$
Moreover,
$$
|A|_{2}^{2}=\int|K(x, y)|^{2} d \mu(x) d \mu(y)
$$