I have this exercise and I don't know how to solve the last question.
In the following $a,b$ are two real numbers such that $a<b$ ,$E=C([a,b],\mathbb{R})$ with the norm $||.||_0$ given by $||.||_0= \sup \lbrace |u(x)|,x\in [a,b] \rbrace)$ and
$E^1=C^1([a,b],\mathbb{R})$ with the norm $||.||_1$ ($||u||_1=||u||_0+||u'||_0$ for $u\in E^1$).
1- Calculate the Green function associated to the problem $-\frac{d^2}{dx^2}$ and the Dirichlet boundary conditions
$u(a)=u(b)=0$ (ok!)2-Determinate eigenvalues and eigenfunctions associated to the problem : $-u''(x)=\lambda u(x) , u(a)=u(b)=0 , x\in (a,b)$ (ok!).
3- Let $L\colon E\rightarrow E^1$ be the operator defined by :
$$Lu(x)=\int_a^b G(x,t) u(t)dt.$$
Prove that $L$ is compact . $L$ is continuous from $E$ to $C^2\cap H^1_0$.
Can someone help me to solve the question 3 ?
Please help me
thank you .