Fredholm alternative theorem

Question

I'm studying in a PDE's course and we have recently used Fredholm alternative theorem as a tool in order to prove the existence and the uniqueness of the solution of a particular problem. We have seen that using the maximum principle it is possible to prove the uniqueness of the solution (assuming that it exists). What I don't get is: why we have uniqueness iff we have the existence?

Answer 1

Fredholm alternative: If $K : X \to X$ is a compact operator, $I + K : X \to X$ is surjective iff it is injective.

In other words, $$(I + X) \, x = 0$$ has only the trivial solution $x = 0$, iff $$(I + X) \, x = b$$ is solvable for every $b$.

That is, uniqueness yields existence (and vice versa).