I am studying for qualifying exams and ran into this problem in Carothers:
Define $T: C[a,b] \rightarrow C[a,b]$ by $(Tf)(x) = \int_a^x f$. Show that $T$ maps bounded sets into equicontinuous (and hence compact) sets.
I have proven that $T(F)$ is uniformly bounded and equicontinuous if $F$ is bounded. However, to use Arzela-Ascoli to show that $T(F)$ is compact I also need that $T(F)$ is closed. I can't see why $T(F)$ is necessarily closed simply given that $F$ is uniformly bounded.
I am going to supply N.S.'s comment as an answer here and accept it, because I think that it is the correct response to the question.
Carothers' claim in parentheses, that T sends bounded sets to compact sets, is incorrect.
Take the subset of $C[0,1]$, $F = \{c | c \in (0,1)\}$. $F$ is clearly uniformly bounded. And $T(F) = \{cx | c \in (0,1)\}$. Consider the sequence $f_n$ in $T(F)$ defined by $f_n(x) = \frac{1}{n+1}x$. $f_n$ has no convergent subsequence. Therefore $T(F)$ is not compact.