  We give a general Lagrange multiplier rule for mathematical programming
problems in a Hausdorff locally convex space. We consider infinitely many
inequality and equality constraints. Our results gives in particular a
generalisation of the result of J. Jahn in \cite{Ja}, replacing
Fr\'echet-differentiability assumptions on the functions by the
Gateaux-differentiability. Moreover, the closed convex cone with a nonempty
interior in the constraints is replaced by a strictly general class of closed
subsets introduced in the paper and called {\it ``admissible sets"}. Examples
illustrating our results are given.
