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====== Fredholm Alternative ======

The Fredholm alternative is a functional analytic generalization of the 'fundamental theorem of linear algebra': for a linear operator $L$ on a finite-dimensional vector space $V$, either the homogeneous equation $Lu = 0$ has a nontrivial solution, or else for each $v \in V$ the inhomogeneous equation $Lu = v$ has a unique solution $u \in V$.

The basis for this theorem is the stability of the index under the addition of compact operators.
I particular, if $X, Y$ are Banach spaces and $D \colon X \to Y$ is Fredholm, then

1. If $K \colon X \to Y$ is compact, then $D + K$ is Fredholm and $D + K, D$ have the same index.

2. There is a constant $\epsilon > 0$ such that the following holds: if $P \colon X \to Y$ has $\| P\| < \epsilon$, then $D + P$ is Fredholm with the same index as $D$.

The Fredholm alternative, as it is usually invoked, takes $aI$ as the Fredholm operator and $-K$ as a compact operator.
Thus if $K$ is compact, $aI - K$ has index zero.
It follows that either both the kernel and cokernel of $aI - K$ are zero, or both the kernel and cokernel of $aI - K$ are nonzero.
Said differently, either $(aI-K)u = 0$ has a nontrivial solution (that is, Ku = au is an eigenvector) or else the cokernel of $I-K$ is non-zero; that is, it is surjective, so $(I-aK)u = v$ has a unique solution for all v.

There is another theorem called the Fredholm alternative: if $L$ is a bounded operator on a Hilbert space, then $Ly = f$ has a solution if and only if $\langle f, z \rangle = 0$ for every $z$ with $L^* z = 0$.
Indeed, the perp to the image of $L$ is the kernel of the adjoint.
Thus if the adjoint is injective, $L$ is surjective. If $L$ is not surjective, the adjoint has some kernel.