Solution
= Solution
The kernel $k(x,y)=e^{-|x-y|}$ is real and symmetric. For $f,g\in L^2[0,1]$, <Fubini's theorem> gives
$$
\langle Kf,g\rangle
=\int_0^1\int_0^1k(x,y)f(y)\overline{g(x)}\,dy\,dx
=\langle f,Kg\rangle,
$$
so $K$ is self-adjoint. It is linear, and because $k\in L^2([0,1]^2)$ it is a <Hilbert-Schmidt operator>, hence bounded, with
$$
\|K\|\leq\|k\|_{L^2([0,1]^2)}<\infty.
$$