Limit superior of subdifferentials of uniformly convergent functions

In this paper we show that the G - subdifferential of a lower semicontinuou s function is contained in the limit superior of the G - subdifferential of lower semicontinuous uniformly convergent family to this function. It happ ens that this result is equivalent to the corresponding normal cones formul as for family of sets which converges in the sense of the bounded Hausdorff distance. These results extend to the infinite dimensional case those of I offe for C-2 - functions and of Benoist for Clarke's normal cone. As an app lication we characterize the subdifferential of any function which is bound ed from below by a negative quadratic form in terms of its Moreau-Yosida pr oximal approximation.