C¹-equivalence of functions near isolated degenerate critical points in Banach space

We prove two theorems of N. Kuiper by standard methods of the calculus on Banach spaces. C⁰-sufficiency for C^r-functions of the r-jet of a real valued C^r-function of lowest degree r₀ ≥ 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 C¹-sufficiency for C^{r + p}-functions, p ≥ r - r₀, 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.