Ordinary Differential Equations and Dynamical Systems Vol2 · Problem 5.9

Linearization and Invariant Manifolds

Mathematics — Calculus · banach-space-differentiability-inverse

Problem

Let

X

and

Y

be Banach spaces. Suppose that

L \in C^1(X, \mathcal{L}(Y, Y))

, and for some

x_0 \in X

L(x_0)

has an inverse in

\mathcal{L}(Y, Y)

. Prove that there is a neighborhood

U \subset X

x_0

on which

L(x)^{-1}

is defined and

L^{-1} \in C^1(U, \mathcal{L}(Y, Y))