ИСТИНА

Let $X$ and $Y$ be Banach spaces, and $Q:X\to Y$ be a continuous quadratic mapping. The following question is under consideration: under what conditions the image of a neighbourhood of zero element in $X$ is a neighbourhood of zero element in $Y$? To say it in other words, under what conditions the (set-valued) right inverse to $Q$ is bounded in some natural sense? For linear mappings the complete answer to this question is given by classical Banach's Open Mapping Theorem: the right inverse to a linear mapping between Banach spaces is bounded if and only if this mapping is onto. We show that for quadratic mappings between infinite-dimensional spaces this is not the case. We give some sufficient conditions for the right-inverse boundedness. We also give the complete solution for the problem under consideration for some particular finite-dimensional cases.