  The representer theorem is one of the most important mathematical foundations
for regularised learning and kernel methods. Classical formulations of the
theorem state sufficient conditions under which a regularisation problem on a
Hilbert space admits a solution in the subspace spanned by the representers of
the data points. This turns the problem into an equivalent optimisation problem
in a finite dimensional space, making it computationally tractable. Moreover,
Banach space methods for learning have been receiving more and more attention.
Considering the representer theorem in Banach spaces is hence of increasing
importance. Recently the question of the necessary condition for a representer
theorem to hold in Hilbert spaces and certain Banach spaces has been
considered. It has been shown that a classical representer theorem cannot exist
in general in non-reflexive Banach spaces. In this paper we propose a notion of
approximate solutions and approximate representer theorem to overcome this
problem. We show that for these notions we can indeed extend the previous
results to obtain a unified theory for the existence of representer theorems in
any general Banach spaces, in particular including $l_1$-type spaces. We give a
precise characterisation when a regulariser admits a classical representer
theorem and when only an approximate representer theorem is possible.
