  The notion of a coherent space is a nonlinear version of the notion of a
complex Euclidean space: The vector space axioms are dropped while the notion
of inner product is kept.
  Coherent spaces provide a setting for the study of geometry in a different
direction than traditional metric, topological, and differential geometry. Just
as it pays to study the properties of manifolds independently of their
embedding into a Euclidean space, so it appears fruitful to study the
properties of coherent spaces independent of their embedding into a Hilbert
space.
  Coherent spaces have close relations to reproducing kernel Hilbert spaces,
Fock spaces, and unitary group representations, and to many other fields of
mathematics, statistics, and physics.
  This paper is the first of a series of papers and defines concepts and basic
theorems about coherent spaces, associated vector spaces, and their topology.
Later papers in the series discuss symmetries of coherent spaces, relations to
homogeneous spaces, the theory of group representations, $C^*$-algebras,
hypergroups, finite geometry, and applications to quantum physics. While the
applications to quantum physics were the main motiviation for developing the
theory, many more applications exist in complex analysis, group theory,
probability theory, statistics, physics, and engineering.
