Abstract
Let E and F be Banach spaces, f: U ⊂ E → F be a map of C r (r ⩾ 1), x 0 ∈ U, and ft (x 0) denote the FréLechet differential of f at x 0. Suppose that f′(x 0) is double split, Rank(f′(x 0)) = ∞, dimN(f′(x 0)) > 0 and codimR(f′(x 0)) s> 0. The rank theorem in advanced calculus asks to answer what properties of f ensure that f(x) is conjugate to f′(x 0) near x 0. We have proved that the conclusion of the theorem is equivalent to one kind of singularities for bounded linear operators, i.e., x 0 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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Ma, J. Complete rank theorem of advanced calculus and singularities of bounded linear operators. Front. Math. China 3, 305–316 (2008). https://doi.org/10.1007/s11464-008-0019-8
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DOI: https://doi.org/10.1007/s11464-008-0019-8