When is there a representer theorem?

Abstract We consider a general regularised interpolation problem for learning parameter vector from data. The well-known representer theorem says that under certain conditions on the regulariser there exists solution in linear span of data points. This is at core kernel methods machine as it makes computationally tractable. Most literature deals only with sufficient theorems Hilbert spaces and shows being norm-based existence theorem. prove necessary reflexive Banach show any has to be essentially exist. Moreover, we illustrate why sense reflexivity minimal requirement function space. further if relies theorem, then independent fact determined by space alone. particular value generalising theory spaces.