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 an m×n real matrix A and a column vector b, either
Ax = b
has a solution or
AT y = 0 has a solution yTb ≠ 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: V → W and you want to solve the equation Tx = b. … If c is an element of W that is not in the image of T, then Tx = c has no solution, by definition. In order for Tx = b to have a solution, the vector b must not have any components in the subspace of W that is complementary to the image of T. This complementary space is the cokernel. The vector b must not have any component in the cokernel if Tx = b is to have a solution.
In this context you could say that the Fredholm alternative boils down to saying either b is in the image of A or it isn’t. If b isn’t in. the image of A, then it has some component in the complement of the image of A, i.e. it has a component in the cokernel, the kernel of AT.
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.
Let K be a compact linear operator on a Hilbert space H. Let I be the identity operator and A = I − K. Let A* denote the adjoint of A.
- The null space of A is finite dimensional,
- The image of A is closed.
- The image of A is the orthogonal complement of the kernel of A*.
- The null space of A is 0 iff the image of A is H.
- The dimension of the kernel of A equals the dimension of the kernel of A*.
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 solution x = 0, in which case the inhomogeneous equation has a unique solution for all f in H.
The other possibility is that homogeneous equation has non-zero solutions, and the inhomogeneous has solutions has a solution if and only if f is orthogonal to the kernel of A*, i.e. if f 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. Requiring f to be orthogonal to the cokernel is a set of constraints.
If the kernel of A has dimension n, then the Fredholm alternative says the cokernel of A also has dimension n.
If solutions x to Ax = f have n degrees of freedom, then right-hand sides f must satisfy n constraints. Each degree of freedom for x corresponds to a basis element for the kernel of A. Each constraint on f corresponds to a basis element for the cokernel that f must be orthogonal to.
[1] Lawrence C. Evans. Partial Differential Equations, 2nd edition