An upper and a lower bound for the distance of a manifold to a nearby point

AbstractA generalization for underdetermined systems of the well-known Newton-Kantorovich theorem gives bounds for the distance of a point, say 0, in Hilbert space X to a nearby manifold S = {x ϵ X ¦ ƒ(x) = 0 }. Here ƒ: X → Y is a differentiable mapping such that Dƒ(0)+ satisfies typical Kantorovich-like conditions. Analysis in the normal space at 0 of S̃ = {x ϵ X ¦ ƒ(x) = ƒ(0)} gives an upper bound of d(0, S). Furthermore the Kantorovich conditions effect S to be locally in a convex cone. The distance of 0 to that cone gives a lower bound of d(0, S)