Im reading Lax book in functional analysis. He proves the Fredholm alternative for compact operators.
I.e
For compact maps $C$ and for $T=I-C$ we have ;
i)$u \in R_{T}$ iff $(u,\ell)=0$ for all $\ell \in N_{T^{'}}$
ii)$dim(N_{T})=dim(N_{T^{'}})$
He does i) by useing spanning criterion and ii) via the identity
$codim(R_{T})-dim(N_{T})=0$ where $T=I-C$, $C$ compact,
which he proved by using the stabilizing property of the nullspace of $T$. He then combined $codim(R_{T})-dim(N_{T})=0$ with the dimension relations of the annihilator $R_{T}$ and the dual of $\frac{X}{R_{T}}$ to get ii)
I can't figure out which oberseration Riesz did in order to use the properties of the transpose/dual to get the results on the spectrum and the relation of dimensions. Lax shows fredholms alternative after the theorem of the properties of the spectrum of compact operators which seems backwards in some sense. To me, w.r.t history, it looks like hilbert uses the fredholm alternative without saying so, and I suppose Rieaz also used Fredholm alternative in his investigation of the spectrum not the following identity $codim(R_{T})-dim(N_{T})=0$. But maybe I got it backwards.
Fredholm was studying integral equations, and he published a landmark paper in 1900 on the subject. Before that people had considered integral equations to be impossibly difficult, and very little was known in general. Fredholm's short 1900 paper "On a new method for the solution of Dirichlet's problem" changed all that. It was Fredholm's paper that inspired Riesz to abstract Fredholm's methods in the form of compact operators.
Fredholm reduced problems of integral equations arising out of partial differential equations to an integral equation "of the second kind" (a term coined by Hilbert to describe Fredholm's equation): $$ \varphi(x) = f(s)+\lambda\int_{a}^{b}K(s,t)f(t)dt $$ Fredholm replaced the integral equation with an approximation coming from looking at Riemann sums. He was then able to write the determinant of the resulting discrete linear system in terms of a well-known expression, and that allowed him to take a limit of the system to obtain the "determinant" of the linear system. Fredholm's solution was a series in the parameter $\lambda$ that he was able to show converged uniformly for $\lambda$ is a bounded subset of the complex plane. He was then able to take the limit of the discrete problems to solve the continuous one. He was able to resolve the integral equation through the use of his "resolvent" kernel (this is where the term "resolvent" of a linear operator originated, and it is Fredholm who is credited with the first definition of a general linear operator.) Fredholm's work produced expressions through determinants, and he looked at the orders of zeros of such expressions. This led to the Fredholm "alternative."
This material can be found in detailed form in J. Dieudonne's book, "A History of Functional Analysis." Dieudonne says this about Fredholm's paper: "This beautiful paper may be considered as the source from which all further developments of spectral theory are derived." There is resolvent, spectrum, Fredholm alternative, and general linear operators found in Fredholm's short paper, along with connections to finite linear systems and compactness results.
In Hilbert's first paper, he studied Fredholm's work, but specialized to symmetric kernels $K(s,t)$, where he was able to obtained much more precise results. He obtained eigenfunctions, Parseval's equality, and introduced what would now be called the unit ball in Hilbert space. Hilbert and his best student Schmidt developed these ideas to where Fredholm's determinants could be eliminated. Hilbert then realized that determinants could be replaced with the older theories of linear equations in orthogonal elements. Hilbert, like Fredholm, was motivated to solve Dirichlet's problem. This is the genesis of Hilbert space.
There was a problem posed to make sense of Fredholm's work in this more general context. F. Riesz solve the problem completely with the introduction of compact operators in his 1918 publication, for which he was able to prove the Fredholm alternative. Dieudonne writes this about Riesz' paper: "In my opinion, F. Riesz's 1918 paper is one of the most beautiful ever written; it is entirely geometric in language and spirit, and so perfectly adapted to its goal that it has never been superseded and that Riesz's proofs can still be transcribed almost verbatim." Very little has changed in how the theory of compact operators is presented today.
Fredholm's work and his use of weakly convergent sequences was subsumed in F. Riesz's elegant definition of compactness; this is the essence of why finite-dimensional kernels and cokernels arise. Co-kernels arise in differential equations as a way of finding linear conditions that will guarantee solvability of $(L-\lambda I)f=g$. The conditions are imposed on $g$, and then the equation is solvable; the kernel, of course, has a natural meaning. The co-kernel conditions are adjoint conditions. Solvability for Fredholm equations requires compability conditions on $g$, and leaves you with constants to be determined because of degeneracy in the $f$. Finite dimensionality of kernels and co-kernels come out of compactness. Equicontinuity of images of smoothing operators, such as inverses of differential operators, forces compactness.