It is well-known that the Lax-Milgram theorem for positive definite quadratic forms on a Hilbert space is useful to show the existence of weak solutions of boundary value problems of elliptic equations. We shall generalize this theorem to the case of a one parameter family of reflexive Banach spaces. We may deal with quadratic forms which are not necessarily positive definite. Some variational inequalities play an essential role for positivity. Then we shall apply our theorem to the construction of weak solutions in to a boundary value problem for an elliptic system.