Generalized Galerkin approximations for pseudoinverses and operator equations of the first kind

The main result of this paper is a basic theorem about generalized Galerkin approximations for pseudoinverses and operator equations of the first kind, which is presented as follows: LetH be a Hilbert space, {H _n} a sequence of closed subspaces ofH,P _n the orthogonal projection ofH ontoH _n,A ∈ ℬ ( H ) andA _n ∈ ℬ ( H_n ). Suppose H _n=H, , ℛ(A _n)=ℛ(A _n) (n ∈ N). Then the following four propositions are equivalent: (a) sup inf ∥v ‖ < ∞ ifu ∈ ℛ (A _n) and ; (b) ; (c) ifu _n ∈ ℛ(A _n) and , thenu∈ℛ(A) and ; (d) ifu _n ∈ ℛ(A _n) and , then u ∈ ℛ(A) and . Furthermore, if any of the above propositions holds, we have thatN(A)= N(A _n), ℛ(A)= ℛ(A _n), ℛ(A)=ℛ(A). Foundation item: Supported by the Wuhan University Teaching Research Foundation (TS2004030) Biography: DU Nailin (1962-), Ph. D., Associate professor, research direction: topology, functional analysis and differential equations.