Compact mapping from $C([0,1])$ to itself.

Question

Let $k \in C([0,1] \times [0,1])$.How can I show that $T_k$ where $$(T_ku)(t)=\int_o^1 k(t,s)u(s)ds,$$ $u \in C([0,1])$, is a compact mapping from $C([0,1])$ to itself?

Answer 1

One possibility is via Arzela-Ascoli.

Another possibility: let $(p_n)$ be a sequence of polynomials in two variables such that $(p_n)$ converges uniformly to $k$ on $[0,1] \times [0,1]$

Define the bounded linear operators $P_n$ by

$(P_nu)(t)=\int_o^1 p_n(t,s)u(s)ds$.

Each $P_n$ is finite - demensional, hence compact. Furthermore:

$||P_n-T_k|| \to 0$ for $n \to \infty$.

Since the set of compact operators is closed, $T_k$ is compact.