Learnability in Hilbert Spaces with Reproducing Kernels

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We explore the question of the learnability of classes of functions contained in a Hilbert space which has a reproducing kernel. We show that if the evaluation functionals are uniformly bounded and if the class is norm bounded then it is learnable. We formulate a learning procedure related to the well known support vector machine (SVM), which requires solving a system of linear equations, rather than the quadratic programming needed for the SVM. As a part of our discussion, we estimate the fat-shattering dimension of the unit ball of the dual of a Banach space when considered as a set of functions on the unit ball of the space itself. Our estimate is based on a geometric property of the Banach space called the type.