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.