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 C^r (r ≥ 1), x₀ ? U, and f′(x₀) denote the Frechet differential of f at x₀. Suppose that f′(x₀) is double split, Rank(f′(x₀)) = ∞, dimN(f′(x₀)) > 0 and codimR(f′(x₀)) > 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.

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