I wanted to show that the integral operator $$T:L_2([0,1])\rightarrow L_2([0,1])$$ defined by $$T(f)(x)=\displaystyle \int _0^1 k(x,t).f(t)dt$$ for all $x\in [0,1]$ where $k:[0,1]\times [0,1]\rightarrow \mathbb{R}$ is any continuous function, is a compact operator.
If $T$ were to be on $C[0,1]$, then we could show it using Arzela-Ascoli theorem. But I couldn’t develop a method for this case.
Because $k$ is continuous you can write it as a uniform limit of two-variable polynomials. This will allow you to write $T$ as a limit of finite-rank operators.