  We characterize the diagonalizable subalgebras of End(V), the full ring of
linear operators on a vector space V over a field, in a manner that directly
generalizes the classical theory of diagonalizable algebras of operators on a
finite-dimensional vector space. Our characterizations are formulated in terms
of a natural topology (the "finite topology") on End(V), which reduces to the
discrete topology in the case where V is finite-dimensional. We further
investigate when two subalgebras of operators can and cannot be simultaneously
diagonalized, as well as the closure of the set of diagonalizable operators
within End(V). Motivated by the classical link between diagonalizability and
semisimplicity, we also give an infinite-dimensional generalization of the
Wedderburn-Artin theorem, providing a number of equivalent characterizations of
left pseudocompact, Jacoboson semisimple rings that parallel various
characterizations of artinian semisimple rings. This theorem unifies a number
of related results in the literature, including the structure of linearly
compact, Jacobson semsimple rings and cosemisimple coalgebras over a field.
