Complete rank theorem of advanced calculus and singularities of bounded linear operators

Abstract

Let E and F be Banach spaces, f: U ⊂ E → F be a map of Cr (r ⩾ 1), x0 ∈ U, and ft (x0) denote the FreLechet differential of f at x0. Suppose that f′(x0) is double split, Rank(f′(x0)) = ∞, dimN(f′(x0)) > 0 and codimR(f′(x0)) s> 0. The rank theorem in advanced calculus asks to answer what properties of f ensure that f(x) is conjugate to f′(x0) near x0. We have proved that the conclusion of the theorem is equivalent to one kind of singularities for bounded linear operators, i.e., x0 is a locally fine point for f′(x) or generalized regular point of f(x); so, a complete rank theorem in advanced calculus is established, i.e., a sufficient and necessary condition such that the conclusion of the theorem to be held is given.