  In this paper we relate the geometry of Banach spaces to the theory of
differential equations, apparently in a new way. We will construct Banach
function space norms arising as weak solutions to ordinary differential
equations of first order. This provides as a special case a new way of defining
varying exponent $L^p$ spaces, different from the Orlicz type approach. We
explain heuristically how the definition of the norm by means of the particular
ODE is justified. The resulting class of spaces includes the classical $L^p$
spaces as a special case. We present an ODE-free means of defining the norms
investigated.
