Let $(X,\|\cdot\|)$ a Banach space. Recall, that a "compact operator" is a map $T:X\longrightarrow X$ such that $T(B)$ is relatively compact, for every bounded subset $B\subset X$. When $X$ is the Banach space of the continuous function $x:[0,1]\longrightarrow \mathbb{R}$, a very important class of compact operators is that defined by certain integral equations (as an immediate consequence of the Arzela-Ascoli theorem), for instance
$$T(x)(t):=\int_{0}^{t}f(t,s,x(s))ds$$
for each $x\in X$.
under suitable conditions of continuity on $f:[0,1]\times [0,1]\times \mathbb{R}\longrightarrow \mathbb{R}$.
So, with $T$ as above, if $B$ is bounded (say, a ball) given $\epsilon>0$ we can "find" a finite set $\{x_{1},\ldots,x_{n}\}\subset T(B)$ such that
$T(B)\subset \{x_{1},\ldots,x_{n}\} + \epsilon B_{X}$,
$B_{X}$ being the closed unit ball of $X$. But, fixed $\epsilon>0$ and $f$, How can we find the above set $\{x_{1},\ldots,x_{n}\}$? Some suggestion?
Thanks!