Proposition 17 ([Uniform compact right inverse]). There are β_0,η_0,C_D>0 such that, for 0<β≤β_0 and 0<T≤η_0β^4 , the operator has a real invariant right inverse D:Y_α→X_α with D≤C_D . No initial value is prescribed for Df .

Define D_ app f=V_i+V_o, R=R_i+J_o. Equations,, and give the exact operator identity D_ app =I+ R, D_ app _ Y_α→X_α ≤C, R_ Y_α→Y_α ≤C(β^2+η^ (1-α)/4 ). Choose β_0 and then η_0 small enough that the last bound is at most 1/2 throughout the asserted range, decreasing the earlier η_0 if necessary. The convergent Neumann series defines D=D_ app (I+ R)^ -1 . It satisfies D=I , is real and invariant, and has uniform norm at most twice the bound for D_ app . This is a right inverse construction and requires no assertion of injectivity.