Operator Compact

Question

I would like a tip for this issue

Let $K : [0, 1] × [0, 1] \rightarrow \mathbb{R}$ be a continuous mapping and $E = (C^{ 0}([0, 1]), ||.||_{\infty})$. Define $J : E \rightarrow E$ by

$$Jf(x) = \int_{0}^{1} K(x, ξ)f(ξ)dξ.$$

Prove that $J$ is compact operator. Does J has an inverse?

thanks