Finite-condimensional Range Inclusion Theorem

The range inclusion theorem was given by Douglas at an article[1]in 1960s.It tells that there will be some equivalent expression if there is range inclusion relationship between operators.The theorem plays a critical role in controllability,in the other words,we can say,for operators,their controlla-bility are equivalent to the inequation.In this article we mainly focus on finite-codimensional range inclusion theorem.Bounded operators A and B on Banach space,we say that A is finite-codimensional included in B,if there is a finite-dimensional subspace M satisfied that Ran(A)C Ran(B)+ M.In this note,we introduce some basic notion and pre-knowledge about range inclusion theorem and state our extension in the first chapter;we give the finite-codimensional theorem in the second chapter.