This question is part of a prelim exam in analysis I'm taking to prepare for my own prelim. I have already proven $K$ is bounded and linear, for an earlier part of the problem.
We are given that $k:[0,1]\times[0,1]\to\mathbb R$ is a fixed, continuous function, and $K: \mathcal C[0,1]\to\mathcal C[0,1]$.
The prompt actually suggests that $$\lVert K \rVert = \sup_{x\in[0,1]}\int_0^1 |k(x,y)|dy$$ and we are asked to prove this. I can show (I think) that the norm is less than or equal to the above, but I cannot seem to show equality. I have tried to come up with an example in which $$\lVert (Kf)(x) \rVert = \lVert f \rVert_\infty \cdot \sup_{x\in[0,1]}\int_0^1 |k(x,y)|dy$$ But I can't do so. My best idea now is to find a sequence of functions $f_n$ that come arbitrarily close to the above equality, but I haven't succeeded there either. Any tips would be much appreciated.
Some ideas: Find $x_0$ such that $\int_0^1 |k(x_0,y)|\,dy$ is a maximum. There is a sequence $f_n$ in $C[0,1]$ that converge pointwise a.e. to $\text { sgn }[k(x_0,y)],$ with $|f_n|\le 1$ for every $n.$ By the dominated convergence theorem,
$$\int_0^1 |k(x_0,y)|\,dy = \int_0^1 k(x_0,y)\text { sgn }[k(x_0,y)]\,dy =\lim_{n\to \infty}\int_0^1 k(x_0,y)f_n(y)\,dy.$$