For 1 less than or equal to p < infinity, the classical L(p) Banach Pr
inciple relates a.e. convergence of a sequence of operators to the fin
iteness of a maximal function, and has had many important applications
. For p = infinity, the classical Banach principle has not been as eff
ective. We give a formulation of an L(infinity) Banach principle which
can be used to establish a.e. convergence (or divergence) of a sequen
ce of operators on L(infinity). (C) 1996 Academic Press, Inc.