  In this brief note we describe relations between the well known notion of a
relatively bounded operator and the operator E-norms considered in
[arXiv:1806.05668].
  We show that the set of all $\sqrt{G}$-bounded operators equipped with the
E-norm induced by a positive operator $G$ is the Banach space of all operators
with finite E-norm and that the $\sqrt{G}$-bound is a continuous seminorm on
this space.
  We also show that the set of all $\sqrt{G}$-infinitesimal operators
(operators with zero $\sqrt{G}$-bound) equipped with the E-norm induced by a
positive operator $G$ is the completion of the algebra $B(H)$ of bounded
operators w.r.t. this norm. Some properties of $\sqrt{G}$-infinitesimal
operators are considered.
