  Echo state networks (ESNs) have been recently proved to be universal
approximants for input/output systems with respect to various $L ^p$-type
criteria. When $1\leq p< \infty$, only $p$-integrability hypotheses need to be
imposed, while in the case $p=\infty$ a uniform boundedness hypotheses on the
inputs is required. This note shows that, in the last case, a universal family
of ESNs can be constructed that contains exclusively elements that have the
echo state and the fading memory properties. This conclusion could not be drawn
with the results and methods available so far in the literature.
