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A few days ago, I found myself needing to use the Fredholm alternative in functional analysis:

Theorem 1 (Fredholm alternative) Let ${X}$ be a Banach space, let ${T: X \rightarrow X}$ be a compact operator, and let ${\lambda \in {\bf C}}$ be non-zero. Then exactly one of the following statements hold:

• (Eigenvalue) There is a non-trivial solution ${x \in X}$ to the equation ${Tx = \lambda x}$.
• (Bounded resolvent) The operator ${T-\lambda}$ has a bounded inverse ${(T-\lambda)^{-1}}$ on ${X}$.

Among other things, the Fredholm alternative can be used to establish the spectral theorem for compact operators. A hypothesis such as compactness is necessary; the shift operator ${U}$ on ${\ell^2({\bf Z})}$, for instance, has no eigenfunctions, but ${U-z}$ is not invertible for any unit complex number ${z}$. The claim is also false when ${\lambda=0}$; consider for instance the multiplication operator ${Tf(n) := \frac{1}{n} f(n)}$ on ${\ell^2({\bf N})}$, which is compact and has no eigenvalue at zero, but is not invertible.

It had been a while since I had studied the spectral theory of compact operators, and I found that I could not immediately reconstruct a proof of the Fredholm alternative from first principles. So I set myself the exercise of doing so. I thought that I had managed to establish the alternative in all cases, but as pointed out in comments, my argument is restricted to the case where the compact operator ${T}$ is approximable, which means that it is the limit of finite rank operators in the uniform topology. Many Banach spaces (and in particular, all Hilbert spaces) have the approximation property that implies (by a result of Grothendieck) that all compact operators on that space are almost finite rank. For instance, if ${X}$ is a Hilbert space, then any compact operator is approximable, because any compact set can be approximated by a finite-dimensional subspace, and in a Hilbert space, the orthogonal projection operator to a subspace is always a contraction. (In more general Banach spaces, finite-dimensional subspaces are still complemented, but the operator norm of the projection can be large.) Unfortunately, there are examples of Banach spaces for which the approximation property fails; the first such examples were discovered by Enflo, and a subsequent paper of by Alexander demonstrated the existence of compact operators in certain Banach spaces that are not approximable.

I also found out that this argument was essentially also discovered independently by by MacCluer-Hull and by Uuye. Nevertheless, I am recording this argument here, together with two more traditional proofs of the Fredholm alternative (based on the Riesz lemma and a continuity argument respectively).