Strong differentiability and the inverse function theorem in Banach spaces

Question

I am trying to prove the strong differentiability version of the Inverse Function Theorem for Banach spaces, but I am not sure if it is true. I am interested in this because it is a kind of punctual version of the theorem. So my main question is:

Is the strong differentiability version of the Inverse Function Theorem true for Banach spaces?

Here is the definition of strong differentiability.

Definition Let $E$ and $E'$ be normed linear spaces, $A \subseteq E$ an open set, $a \in A$ a point and $f: A \to E'$ a function. We say $f$ is strong differentiable at $a$ when there is a continuous linear map $D: E \to E'$ such that $$\lim_{(x,x') \to (a,a)} \frac{f(a+x')-f(a+x) - D(x'-x)}{x'-x} = 0.$$

In this case, $f$ is differentiable at $a$ and $D = Df|_a$, that is, the linear map $D$ is the differential of $f$ at $a$.

Considerer the remainder function $r_a(v) = f(a+v) - f(a) - Df|_a(v)$. In finite dimensional spaces, strong differentiability at $a$ can be shown to be equivalent to this: for every $\varepsilon > 0$, there is a neighborhood of the origin in which the function $r_a$ is Lipschitz with Lipscitz constant $\varepsilon$. I believe this is also true for infinite dimensions, but have not proved it yet.

Inverse Function Theorem (strongly differentiable) Let $E$ and $E'$ be Banach spaces, $A \subseteq E$ an open set, $a \in A$ a point and $f: A \to E'$ a function which is strongly differentiable at $a$ and such that $Df|_a:E \to E'$ is a linear isomorphism. In this case, there is an open neighborhood $V \subseteq A$ of $a$ such that $f|_V: V \to f(V)$ is a homeomorphism, the inverse function $f^{-1}: f(V) \to V$ is strongly differentiable at $f(a)$ and its differentiable at $f(a)$ is $Df^{-1}|_{f(a)} = (Df|_{a})^{-1}$.

Answer 1

Yes, it is true. This inverse function theorem is in a sense half-way between the Lipschitz and the $C^1$ setting. To be more precise let me review some classic results.

1. (Invertibility of Lipschitz perturbations of the identity). Let $(E,\|\cdot\|)$ be a Banach space; $A\subset E$ an open set, $I:A\to E$ the inclusion map, $h:A\to E$ a $\lambda$-Lipschitz map, with $\lambda<1$, and $f:=I+h$. Then

   • i) The set $f(A)$ is open in $E$, precisely, if $ B(a,r)\subset A$, then $B(f(a),(1-\lambda)r)\subset f(B(a,r))$;
   • ii) $f:A\to f(A)$ is a bi-Lipschitz homeomorphism; precisely $f^{-1}$ is $\frac 1 {1-\lambda}$-Lipschitz and $f^{-1}=I+k$ with $k=f^{-1}-I=-h\circ f^{-1}$, a $\frac \lambda {1-\lambda}$-Lipschitz map.

   (The less immediate part is i, which uses the contraction principle to prove the local surjectivity; part ii comes from elementary inequalities).

2. The above fact generalizes immediately to Lipschitz perturbations $$f:=T+h=T(I+T^{-1}h)$$ of an invertible linear operator $T\in L(E,E')$ between Banach spaces, now with the condition $\lambda<\|T^{-1}\|^{-1}$ on the Lipschitz constant $\lambda$ of $h$, so that (1) applies. If we bother to write the corresponding Lipschitz estimates, we have that $ f^{-1}=(I+T^{-1}h)^{-1}T^{-1} $ is Lipschitz of constant $\frac{\|T^{-1}\|}{1-\lambda\|T^{-1}\|},$ and we may write $f^{-1}= T^{-1}+k$ with $k:=f^{-1}-T^{-1}=-T^{-1}hf^{-1}$, thus a map with Lipschitz constant $\frac{\lambda\|T^{-1}\|^2}{1-\lambda\|T^{-1}\|}.$

3. If in the latter we don't want to make assumptions on $\|T^{-1}\|$, but we are happy with a local invertibility at a point $a\in A$, the natural assumption is that of the $C^1$ local inversion theorem: $f\in C^1(A,E')$ with invertible $Df(a):=T$. And if we prefer to stay in a Lipschitz setting and not to assuming $C^1$: The map $f$ writes $f=T+h$ with arbitrarily small $\text{lip}(h)$ on sufficiently small nbd $U$ of $a$ (which is exactly your "strong differentiability condition"), so that in some sufficiently small nbd of $a$ one has $\text{lip}(h)<\|T^{-1}\|^{-1}$ as required in (2). Note that in this assumption, by the Lipschitz estimates in (2), one also has $f^{-1}=T^{-1}+k$, with arbitrarily small $\text{lip}(k)$ on sufficiently small nbd of $b:=f(a)$, that is, $Df^{-1}(b)=T^{-1}$ as "strong differential".

Answer 2

I think this concept and the associated inverse and implicit function theorems are covered in Glöckner's "Implicit functions from topological Vector spaces to Banach spaces". The arxiv Version of this paper is here.