We propose a framework for dealing with binary hard-margin
classification in non-Euclidean Banach spaces, centering on the use of a
supporting semi-inner-product taking the place of an
inner-product in Hilbert spaces.  The theory of semi-inner-product
spaces allows for a geometric, Hilbert-like formulation of the
problems, and we show that a surprising number of results from the
Euclidean case can be appropriately generalised.  These include the
Representer theorem, convexity of the associated optimization
programs, and even, for a particular class of Banach spaces, a
``kernel trick'' for non-linear classification.  Statistical properties
of the hyperplanes are discussed, as well as applications to 
learning in metric spaces.