Definition 1

Let $X$ be a real Banach space and $X^+\subset X$.

Then, $X^+$ is called a quadrant of $X$ iff there exists a nonempty finite linearly independent subset $\{\lambda_1,...,\lambda_n\}$ of continuous linear functionals on $X$ such that $X^+=\{x\in X: \forall 1\leq i \leq n, \lambda_i(x)\geq 0\}$.


Definition 2

Let $X,Y$ be Banach spaces and $E$ be a subset of $X$ and $p\in E$ be a limit point of $E$.

Let $f:E\rightarrow W$ be a function.

Then, $f$ is said to be (Fréchet) differentiable at $p$ iff there exists a continuous linear transformation $T:X\rightarrow Y$ such that $\lim_{x\to p} \frac{||f(x)-f(p)-T(x-p)||}{||x-p||} =0$.

(Note that differentiability can be defined at a point which is a limit point of the domain of the function)

Let $X,Y$ be real Banach spaces and $X^+,Y^+$ be quadrants of of $X,Y$ respectively, and $E$ be open in $X^+$.

Let $f:E\rightarrow Y^+$ be a (Fréchet) $C^1$-function and let $a\in E$ such that $Df(a)$ is invertible.

Then, how do I prove that there are open sets $V$ and $W$ in $X^+$ and $Y^+$ respectively such that $a\in V\subset E$ and $f(V)=W$ and $f\upharpoonright V: V\rightarrow W$ is a diffeomorphism?

I know how to prove the inverse function theorem for functions whose domains are open in real or complex Banach spaces. However, all those proofs I know depend on Banach contraction principle, and I have a trouble to apply the contraction principle in the given case above. I don't know whether the statement is true, but I certainly believe that it must be true.

I think my previous posts should be used to prove this, and belows are the lists:

  1. How do I show that for linearly independent set in dual is a dual of a linearly independent set?

  2. Show that quadrants are homeomorphic

Thank you in advance.


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