an impressive method for adjoint of linear and nonlinear operators

In this paper, we have obtained the adjoint of an arbitrary operator (linear and nonlinear) in hilbert space by introducing an n-dimensional riemannian manifold. this general formalism covers every linear operator (non-differential) in hilbert space. in fact, our approach shows that instead of directly using an operator's adjoint definition, it can be obtained directly by relying on a suitable generalized space according to the action of the operator in question. in the case of nonlinear operators, we must change the definition of the linear operator adjoint. however, here, we have obtained an adjoint of these operators concerning the definition of the derivative of the operator. we have shown one of the straight applications of ''frechet derivative'' in the algebra of the operators.