  For locally convex vector spaces (l.c.v.s.) $E$ and $F$ and for linear and
continuous operator $T: E \rightarrow F$ and for an absolutely convex
neighborhood $V$ of zero in $F$, a bounded subset $B$ of $E$ is said to be
$T$-V-dentable (respectively, $T$-V-s-dentable, respectively, $T$-V-f-dentable)
if for any $\epsilon>0$ there exists an $x\in B$ so that $ x\notin
\overline{co} (B\setminus T^{-1}(T(x)+\epsilon V))$ (respectively, so that $
x\notin s$-$co (B\setminus T^{-1}(T(x)+\epsilon V)),$ respectively, so that $
x\notin {co} (B\setminus T^{-1}(T(x)+\epsilon V)) ). $ Moreover, $B$ is called
$T$-dentable (respectively, $T$-s-dentable, $T$-f-dentable) if it is
$T$-V-dentable (respectively, $T$-V-s-dentable, $T$-V-f-dentable) for every
absolutely convex neighborhood $V$ of zero in $F.$ RN-operators between locally
convex vector spaces have been introduced in [5]. We present a theorem which
says that, for a large class of l.c.v.s. $E, F,$ if $T: E \rightarrow F$ is a
linear continuous map, then the following are equivalent: 1) $T \in RN(E,F);$
2) Each bounded set in $E$ is $T$-dentable; 3) Each bounded set in $E$ is
$T$-s-dentable; 4) Each bounded set in $E$ is $T$-$f$-dentable. Therefore, we
have a generalization of Theorem 1 in [8], which gave a geometric
characterization of RN-operators between Banach spaces.
