Abstract
We prove two theorems of N. Kuiper by standard methods of the calculus on Banach spaces. C 0-sufficiency for C r -functions of the r-jet of a real valued C r -function of lowest degree r 0 ⩾ 2 on a Hilbert space follows from Kuiper's condition Q( r) also on a Banach space endowed with smooth partitions of unity. A new proof is given of the C 1-sufficiency for C r + p -functions, p ⩾ r − r 0, of the r + p-jet of a Q(r)-function on Hilbert space that also holds on Banach space modulo a further condition on the behavior of the function near its degenerate critical point.
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