Fredholm Alternative

The Fredholm alternative is so called because it is a theorem by the Swedish mathematician Erik Ivar Fredholm that has two alternative conclusions: either this is true or that is true. This post will state a couple forms of the Fredholm alternative.

Mr. Fredholm was interested in the solutions to linear integral equations, but his results can be framed more generally as statements about solutions to linear equations.

This is the third in a series of posts, starting with a post on kernels and cokernels, followed by a post on the Fredholm index.

Fredholm alternative warmup

Given anm*n real matrix A and a column vectorb, either

Ax =b

has a solution or

A^T y = 0 has a solutiony^Tb 0.

This is essentially what I said in an earlier post on kernels and cokernels. From that post:

Suppose you have a linear transformation T:VWand you want to solve the equationTx=b. ... If cis an element ofWthat is not in the image ofT, thenTx=chas no solution, by definition. In order forTx=bto have a solution, the vectorbmust not have any components in the subspace ofWthat is complementary to the image ofT. This complementary space is the cokernel. The vectorbmust not have any component in the cokernel ifTx=bis to have a solution.

In this context you could say that the Fredholm alternative boils down to saying either b is in the image ofA or it isn't. Ifb isn't in. the image ofA, then it has some component in the complement of the image ofA, i.e. it has a component in the cokernel, the kernel of A^T.

The Fredholm alternative

I've seen the Fredholm alternative stated several ways, and the following from [1] is the clearest. The alternative" nature of the theorem is a corollary rather than being explicit in the theorem.

As stated above, Fredholm's interest was in integral equations. These equations can be cast as operators on Hilbert spaces.

LetK be a compact linear operator on a Hilbert space H. LetI be the identity operator andA = I - K. LetA* denote the adjoint ofA.

1. The null space of Ais finite dimensional,
2. The image of A is closed.
3. The image ofA is the orthogonal complement of the kernel ofA*.
4. The null space ofA is 0 iff the image ofA isH.
5. The dimension of the kernel ofA equals the dimension of the kernel ofA*.

The last point says that the kernel and cokernel have the same dimension, and the first point says these dimensions are finite. In other words, the Fredholm index of A is 0.

Where is the alternative" in this theorem?

The theorem says that there are two possibilities regarding the inhomogeneous equation

Ax = f.

One possibility is that the homogeneous equation

Ax = 0

has only the solutionx = 0, in which case the inhomogeneous equation has a unique solution for allf inH.

The other possibility is that homogeneous equation has non-zero solutions, and the inhomogeneous has solutions has a solution if and only iff is orthogonal to the kernel ofA*, i.e. iff is orthogonal to the cokernel.

Freedom and constraint

We said in the post on kernels and cokernels that kernels represent degrees of freedom and cokernels represent constraints. We can add elements of the kernel to a solution and still have a solution. Requiringf to be orthogonal to the cokernel is a set of constraints.

If the kernel ofA has dimensionn, then the Fredholm alternative says the cokernel ofA also has dimensionn.

If solutions x to Ax =f haven degrees of freedom, then right-hand sidesf must satisfyn constraints. Each degree of freedom forx corresponds to a basis element for the kernel ofA. Each constraint onf corresponds to a basis element for the cokernel thatf must be orthogonal to.

[1] Lawrence C. Evans. Partial Differential Equations, 2nd edition

The post Fredholm Alternative first appeared on John D. Cook.