Motivation (skip if you want). I am reading a classic paper by Donaldson about orientation of Yang-Mills moduli space https://projecteuclid.org/euclid.jdg/1214441485 I am having an hard time in understanding the proof of Lemma 3.9 I'll abstract the setup for you.
Setup. Let $A:X\to Y$ be a bounded Fredholm operator between Hilbert spaces. Suppose that $$||A^*y||\geq \frac 1 2 ||y||$$ (A is the operator $\hat{D}_u\oplus \hat{S}$ in Donaldson's paper) so that $A$ is surjective. Let $V<X$ be a subspace such that $$||A v||<\frac 1 2 ||v||\quad \quad \forall v\in V$$
This should imply that the projection onto the kernel of $A$, $\pi:X\to \ker A$ is injective when restricted to $V$. How can we prove that $\pi|_V$ is injective? Or equivalently $$V\cap \ker A^\perp = V\cap Im(A^*)=\emptyset.$$
Suppose $v \in V\setminus\{0\}$ has $v = A^*y$. Then, since $||v|| \ge \frac{1}{2}||y||$ and $||Av|| < \frac{1}{2}||v||$, $$||v||^2 = \langle A^*y, v\rangle = \langle y, Av\rangle \le ||y|| \text{ } ||Av|| < 2||v|| \frac{1}{2}||v||,$$ a contradiction.