High-order open mapping theorems
نویسنده
چکیده
The well-known finite-dimensional first-order open mapping theorem says that a continuous map with a finite-dimensional target is open at a point if its differential at that point exists and is surjective. An identical result, due to Graves, is true when the target is infinite-dimensional, if “differentiability” is replaced by “strict differentiability.” We prove general theorems in which the linear approximations involved in the usual concept of differentiability is replaced by an approximation by a map which is homogeneous relative to a one-parameter group of dilations, and the error bound in the approximation involves a “homogeneous pseudonorm” or a “homogeneous pseudodistance,” rather than the ordinary norm. We outline how these results can be used to derive sufficient conditions for openness involving higher-order derivatives, and carry this out in detail for the second-order case. ∗Supported in part by NSF Grant DMS01-03901.
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