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Random Talk on Random Thoughts

Injectivity of Stable Mappings

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Two weeks ago, I revised calculus and read the proof of the Inverse Function Theorem in the calculus textbook that I was reading.1 The Nonlinear Stablity Theorem is first applied to a continuously differentiable function $\vect{F}: \mathcal{O} \to \R^n$, where $\mathcal{O}$ is an open subset of $\R^n$ and contains a point $\vect{x}_*$ at which $\det \vect{DF}(\vect{x}_*) \ne 0$, to construct a neighbourhood $U$ of $\vect{x}_*$ and find $c > 0$ such that $\norm{\vect{F}(\vect{u})-\vect{F}(\vect{v})} \ge c \norm{\vect{u} - \vect{v}}\quad\forall\,\vect{u},\vect{v} \in U$.

It’s said that if $V := \vect{F}(U)$, then $\vect{F}: U \to V$ is bijective. The surjectivity is obvious, but I wasn’t smart enough to see the injectivity immediately. After a week, I realised that I overlooked the discussion about invertible linear operators on $\F^n$ that preceeded the introduction to the idea of stable mapping. In fact, $\forall\,\vect{u},\vect{v} \in U,\vect{F}(\vect{u}) = \vect{F}(\vect{v}) \iff \norm{\vect{F}(\vect{u}) - \vect{F}(\vect{v})} = 0$.

Then $\norm{\vect{u} - \vect{v}} \le \norm{\vect{F}(\vect{u}) - \vect{F}(\vect{v})} / c = 0 \iff \vect{u} = \vect{v}$.

  1. FitzPatrick’s Advanced Calculus