  We consider a general regularised interpolation problem for learning a
parameter vector from data. The well known representer theorem says that under
certain conditions on the regulariser there exists a solution in the linear
span of the data points. This is the core of kernel methods in machine learning
as it makes the problem computationally tractable. Necessary and sufficient
conditions for differentiable regularisers on Hilbert spaces to admit a
representer theorem have been proved. We extend those results to
nondifferentiable regularisers on uniformly convex and uniformly smooth Banach
spaces. This gives a (more) complete answer to the question when there is a
representer theorem. We then note that for regularised interpolation in fact
the solution is determined by the function space alone and independent of the
regulariser, making the extension to Banach spaces even more valuable.
