Consider $T: C([0,1]) \rightarrow C([0,1])$ defined by $$(Tf)(t) := \int_0^1 \kappa_t(s)f(s)ds,$$ where $\kappa:[0,1]^2 \rightarrow \mathbb{R}$ satisfies the following properties:
- for all $t\in [0,1]$, the function $\kappa_t(s)$ is integrable in $s$;
- $t\rightarrow \kappa_t(s) \in L^1([0,1])$ is continuous.
Show that $T$ is a compact operator.
My proof: Let $f_n$ be a bounded sequence in $C([0,1])$ with the $\sup$ norm, we show that the sequence satisfies the assumption of Arzelà–Ascoli theorem, which would imply that $T$ is a compact operator.
To show $(Tf_n)(t)$ is uniformly bounded. By Schwartz inequality, we have $$|(Tf_n)(t)| \leq ||\kappa_t||_1 ||f_n ||_\infty,$$ the quantity $||f_n||_\infty$ is bounded for each $n$ by assumption, and $||\kappa_t||_1$ is bounded over $t\in [0,1]$ because Property 2.
To show $(Tf_n)(t)$ is equicontinuous, let $t_i \rightarrow t$ $$\lim_{i\rightarrow \infty}\bigg|(Tf_n)(t_i) - (Tf_n)(t)\bigg|\leq \lim_{i\rightarrow \infty}\int_0^1 |\kappa_{t_i}(s) - \kappa_t(s)|\cdot |f_n(s)|ds \leq \lim_{i\rightarrow \infty}||\kappa_{t_i} - \kappa_t||_1 ||f_n ||_\infty,$$ which goes to zero (because of Property 2) independent of $n$ since $||f_n ||_\infty$ is bounded.
This is a problem from a Ph.D. entrance exam, I am a little unsure because my proof seemed very short compared to other problems in the set. Is this correct? Thank you very much for reading.
There is no problem for boundedness.
However, for equi-continuity, you have to be more careful. First of all, we should show that $\lim_{i\to\infty}\sup_n\dots\to 0$ (there won't be any problem because the estimates are uniform in $n$, but we have to take the supremum). Second, this proves the equi-continuity at $t$ and we want a uniform equi-continuity. This can be done using the suggested estimate, namely, $$\sup_n|T(f_n)(s)-T(f_n)(t)|\leqslant \lVert\kappa_s-\kappa_t\rVert_1$$ and using the unifom continuity of $s\mapsto \kappa_s$ in $L^1$.