A note on a scale-sensitive dimension of linear bounded functionals in banach spaces

Abstract

We show that "B is of type p > 1" is a necessary and sufficient condition for a learnability of a class of linear bounded functionals with norm ≤ I restrited to the unit ball in Banach Space B. On the way we give very short probabilistic proof for Vapnik's result (Hilbert Space and improved) and improve our result with Pascal Koiran for convex halls of indicator functions. The approach we use in this paper allows to connect various results about learnability and approximation.