The best generalized inverse of the linear operator in normed linear space

Let X,Y be normed linear spaces, T∈L(X,Y) be a bounded linear operator from X to Y. One wants to solve the linear problem Ax=y for x (given y∈Y), as well as one can. When A is invertible, the unique solution is x=A-1y. If this is not the case, one seeks an approximate solution of the form x=By, where B is an operator from Y to X. Such B is called a generalised inverse of A. Unfortunately, in general normed linear spaces, such an approximate solution depends nonlinearly on y. We introduce the concept of bounded quasi-linear generalised inverse Th of T, which contains the single-valued metric generalised inverse TM and the continuous linear projector generalised inverse T+. If X and Y are reflexive, we prove that the set of all bounded quasi-linear generalised inverses of T, denoted by GH(T), is not empty In the normed linear space of all bounded homogeneous operators, the best bounded quasi-linear generalised inverse Th of T is just the Moore–Penrose metric generalised inverse TM. In the case, X and Y are finite dimension spaces Rn and Rm, respectively, the results deduce the main result by G.R. Goldstein and J.A. Goldstein in 2000.