this was a question given in a list of many given by our teacher. It states:
If $k : [0, 1] \times [0, 1] \rightarrow K$ is continuous. Then we have to prove that the operator $T : L^1([0, 1]) \rightarrow C^0([0, 1])$ defined by $$Tf(x) :=\int_{ [0,1]}k(x, y)f(y) dy; $$ for $f \in L^1([0, 1]), x \in [0, 1]$ is well-defined and compact.
Our teacher gave us a hint that we need to use the Arzela-Ascoli theorem and that we should observe that uniform continuity of $k$ is useful.
Could someone please help me out? Thank you.