  The goal of this paper is to generalize the theory of triangularizing
matrices to linear transformations of an arbitrary vector space, without
placing any restrictions on the dimension of the space or on the base field. We
define a transformation T of a vector space V to be "triangularizable" if V has
a well-ordered basis such that T sends each vector in that basis to the
subspace spanned by basis vectors no greater than it. We then show that the
following conditions (among others) are equivalent: (1) T is triangularizable,
(2) every finite-dimensional subspace of V is annihilated by f(T) for some
polynomial f that factors into linear terms, (3) there is a maximal
well-ordered set of subspaces of V that are invariant under T, (4) T can be put
into a crude version of the Jordan canonical form. We also show that any finite
collection of commuting triangularizable transformations is simultaneously
triangularizable, we describe the closure of the set of triangularizable
transformations in the standard topology on the algebra of all transformations
of V , and we extend to transformations that satisfy a polynomial the classical
fact that the double-centralizer of a matrix is the algebra generated by that
matrix.
