On $U$-Dominant Dimension
  Let $\Lambda$ and $\Gamma$ be artin algebras and $_{\Lambda}U_{\Gamma}$ a faithfully balanced selforthogonal bimodule. We show that the $U$-dominant dimensions of $_{\Lambda}U$ and $U_{\Gamma}$ are identical. As applications to the results obtained, we give some characterizations of double dual functors (with respect to $_{\Lambda}U_{\Gamma}$) preserving monomorphisms and being left exact respectively. 
Zhaoyong Huang