Multiparameter eigenvalue problems (Enhanced Edition‪)‬

The theory of linear spaces draws its inspiration from sources ranging
from operations on vectors in euclidean space, the theory of linear equations
in a finite number of unknowns, through Fourier series and the theory of
equations in an infinity of unknowns and specific function-spaces such as the
LP-spaces, to abstract spaces such as those associated with the names of
Hilbert and Banach. In the most general notion of all, we assume nothing
more than that we have a collection of unspecified objects that admit linear
operations, subject to certain formal axioms such as the commutative and
distributive laws. This constitutes the basis of the abstract or “algebraic”
theory of linear spaces. Although its meager foundations leave little room
for analytical subtleties, certain formal constructions can be made, and some
formal results proved with the advantages of generality and compact formulation.
It is the latter advantage that is significant in what follows. We shall
consider mainly finite-dimensional spaces, and so will really be dealing with
matrix algebra, but have much need to consider constructions involving two
or more spaces.