  The authors lay the foundations for the study of normal families of
holomorphic functions and mappings on an infinite-dimensional normed linear
space. Characterizations of normal families, in terms of value distribution,
spherical derivatives, and other geometric properties are derived. Montel-type
theorems are established.
  A number of different topologies on spaces of holomorphic mappings are
considered. Theorems about normal families are formulated and proved in the
language of these various topologies.
  Normal functions are also introduced. Characterizations in terms of
automorphisms and also in terms of invariant derivatives are presented.
