Fredholmin

Fredholmin is a term that can refer to several related concepts within mathematics, primarily in the field of integral equations and functional analysis. The most common association is with the Fredholm alternative, a fundamental theorem concerning the existence and uniqueness of solutions to Fredholm integral equations. This theorem, named after the Swedish mathematician Erik Ivar Fredholm, states that a Fredholm integral equation of the second kind either has a unique solution for every right-hand side, or it has a non-trivial homogeneous solution, in which case there are infinitely many solutions for certain right-hand sides and no solutions for others.

The theory of Fredholm operators is a generalization of this concept to a broader class of linear operators in functional analysis. A Fredholm operator is a bounded linear operator between Banach spaces whose kernel is finite-dimensional and whose image is closed and has a finite-dimensional cokernel. The Fredholm index of such an operator is defined as the difference between the dimension of its kernel and the dimension of its cokernel. This concept is crucial in the study of differential equations and their generalizations, as it provides a way to analyze the solvability of boundary value problems.

Fredholm determinants are also an important related concept. For certain operators, particularly compact operators, a Fredholm determinant can be defined, which is a function that captures information about the eigenvalues of the operator. This is closely linked to the Fredholm resolvent, which helps in solving Fredholm integral equations. In essence, Fredholmin encompasses a suite of mathematical tools and theorems developed to understand and solve a significant class of linear equations and operators.