Recently, E. Pap showed that the adjoint operator $T'$ for any linear operator $T$ with the domain being a normed $K$-space is bounded. E. Pap and C. Swartz proved a locally convex version of this Adjoint Theorem. In this paper a generalization is given of the Adjoint Theorem on operators with the domain being a locally convex $A$-space, which was introduced by R. Li and C. Swartz. The obtained results are applied to derive a version of the Closed Graph Theorem. Some limitations for further generalizations of the Closed Graph Theorem and Banach-Steinhaus Theorem with respect to infrabarrelledness are pointed out.