Let $X=(-1,1)^{p}$. I am given a differentiable map $f:X\to\mathbb{R}^{n}$, with $n\gg p$, that factors as $$f(x)=PF(x)$$ where $F:X\to H$ is a smooth bounded injective map onto some Hilbert space $(H,\|\cdot\|)$, while $P:H\to\mathbb{R}^{n}$ is a linear operator. Assume that:
i) $F$ admits a continuous inverse $F^{-1}: F(X) \to (-1,1)^{p}$;
ii) the Jacobian of $f$ is always of full rank, i.e. $\text{rank}(J_{f}(x))=p$ for all $x\in X.$
Does it follow that $f$ is injective?
Side note: I wanted to apply Hadamard's Global Inverse Theorem, but I don't think that $\mathcal{S}:=f(X)\subset\mathbb{R}^{n}$ can be considered a smooth $p$-manifold. Furthermore, since $(-1,1)^{p}$ is open, I think $f$ might fail to be proper (to this end: I would also be satisfied if we could show that $f$ is injective on $[-1+\varepsilon, 1-\varepsilon]^{p}$ for arbitrary $\varepsilon>0$).