Much research has been done on the Inexact Newton-like methods for solving nonlinear equations and unconstrained optimization problems.However, no result on inexact rank one and rank two updated quasi-newton methods has been seen , maybe because rank one and rank two updated quasi-newton equations are easier to solve than Newton equations. When Quasi-Newton methods is extended to operator equations in Hilbert space or minimax problems , it is difficult or impossible to solve subproblems exactly . so it is necessary to investigate inexact quasi-newton method in Hilbert space. In [1] the author has investigated the convergence property of inexact quasi-newton method for nonlinear equations in finite dimensional space. It has been prepared for the investigation of inexact quasi-newton method for operator equations in infinite dimensional space.On the basis of [1], we extend inexact quasi-newton method to infinite dimensional space. For the original infinite dimensional problem, if the inexact quasi-newton method with Broyden updating can not converge quickly , for example, it can not attain super linear convergence rate , then when the discretization is becoming finer and finer , we can not get fast convergence rate for the corresponding finite dimensional problem. So we need to study the convergence problem in infinite dimensional spaces. In this paper , we study the convergence property of inexact quasi-newton method in infinite dimensional Hilbert space. It's the basis for solving operator equations in Hilbert space with inexact quasi-newton method and projection method.
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