In this note the impossibility of weak convergence of the density esti
mator in L-2(R) is established. On the other hand a class of non-trivi
al transforms of the estimator is specified that do yield a Gaussian l
imiting measure on L-2(R). As a corollary we obtain a result for weak
convergence of certain quadratic functionals that in a sense complemen
ts a result by Bickel and Rosenblatt (1973).