Lemma 2

Let $x$ be a point of X$,\varphi : X\rightarrow Y$ a morphism of class $C^1$ that is an immersion at $x$. There exists a neighbourhood $\Omega$ of $\varphi$ in $\mathscr{C}^1(X;Y)$ and a neighbourhood B of $x$ in X such that, for all $\psi \in \Omega$, the restriction of $\psi$ to B is injective.

Let U be a relatively compact open neighbourhood of $x$ isomorphic to a finite dimensional vector space, and such that $\varphi (U)$ is contained in the domain V of a chart. The set $\Omega_0$ of $\psi \in \mathscr{C}^1(X;Y)$ such that $\psi (U)\subset V$ is open in $\mathscr{C}^1$(X;Y), and the restriction map $\Omega_0\rightarrow \mathscr{C}^1(U;V)$ is continuous; we are thus reduced to proving the lemma when X = U and Y = V, in other words, we can assume that X is a finite dimensional vector space and Y is a Banach space. Choose norms on X and Y.

The linear map $D\varphi (x) : X\rightarrow$ Y is injective; denote its conorm by $q($Spectral Theory, Chap. III, §2, no. 6), so that, by definition, we have $\|D\varphi (x).t\| \geq q\|t\|$ for all $t\in X$. Let $\varepsilon \in \mathbf{R}$ be such that $0< \varepsilon < q/2$, and let B be a closed ball with centre $x$ such that $\|D\varphi (u)-D\varphi (x)\| \leq \varepsilon$ for all $u\in B$. Denote by $\Omega$ the subset of $\mathscr{C}^1(X;Y)$ consisting of the morphisms $\psi$ such that $\|D\psi (u)-D\varphi (u)\| \leq \varepsilon$ for all $u\in B$; it is open by definition of the topology of $\mathscr{C}^1(X;Y)$. For $\psi \in \Omega$, put $\psi_0=\psi -D\varphi (x)$. We have $\|D\psi_0(u)\| \leq 2\varepsilon$ for all $u\in B$, and consequently $\|\psi_0(u)-\psi_0(v)\| \leq 2\varepsilon \|u-v\|$ for all $u$ and $v$ in B (Differentiable and Analytic Manifolds, Results, 2.2.3). It follows that

Consequently, the restriction of $\psi$ to B is injective, hence the lemma.