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Abstract

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 (ODE) of the first order. This provides as a special case a new way of defining varying exponent $L^p$ spaces, different from the Musielak–Orlicz type approach. We explain heuristically how the definition of the norm by means of a particular ODE is justified. The resulting class of spaces includes the classical $L^p$ spaces as a special case. A noteworthy detail regarding our $L^{p(\cdot )}$ norms is that they satisfy Hölder’s inequality (properly).