For a problem I'm working on I have two Banach spaces $X, Y$ and an injective immersion $T\colon X \to Y$ (that is, a $C^1$ injective mapping having the property that its (Fréchet) differential $dT (x)$ is injective at any $x \in X$).
I'm mainly interested in the restriction of $T$ to a finite-dimensional subspace $V_n\subset X$. Can I conclude that $T(V_n)\subset Y$ is contained in a subspace of $Y$ having the same dimension as $V_n$?
Thank you.