  A general form of the Lions-Magenes theorems on solvability of an elliptic
boundary-value problem in the spaces of nonregular distributions is proved. We
find a general condition on the space of right-hand sides of the elliptic
equation under which the operator of the problem is bounded and has a finite
index on the corresponding couple of Hilbert spaces. Extensive classes of the
spaces satisfying this condition are constructed. They contain the spaces used
by Lions and Magenes.
