  We study the 'no-dimension' analogue of Carath{\'e}odory's theorem in Banach
spaces. We prove such a result together with its colorful version for uniformly
smooth Banach spaces. It follows that uniform smoothness leads to a greedy
de-randomization of Maurey's classical lemma \cite{pisier1980remarques}, which
is itself a 'no-dimension' analogue of Carath{\'e}odory's theorem with a
probabilistic proof. We find the asymptotically tight upper bound on the
deviation of the convex hull from the $k$-convex hull of a bounded set in $L_p$
with $1 < p \leq 2$ and get asymptotically the same bound as in Maurey's lemma
for $L_p$ with $1 < p < \infty.$
