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In this paper, we present a subspace method for solving large scale nonlinear equality constrained optimization problems. The proposed method is based on a SQP method combined with the limited-memory BFGS update formula. Each subproblem is solved in a theoretically suitable subspace. In the case of few constraints, we show that our search direction in the subspace is equivalent to that of the SQP subproblem in the full space. In the case of many constraints, we reduce the number of constraints in the subproblem and we show that the solution of the subspace subproblem is a descent direction of a particular exact penalty function. Global convergence properties of the proposed method are given for both cases. Numerical results are given to illustrate the soundness of the proposed model.