  The Statistical Learning Theory (SLT) provides the theoretical guarantees for
supervised machine learning based on the Empirical Risk Minimization Principle
(ERMP). Such principle defines an upper bound to ensure the uniform convergence
of the empirical risk Remp(f), i.e., the error measured on a given data sample,
to the expected value of risk R(f) (a.k.a. actual risk), which depends on the
Joint Probability Distribution P(X x Y) mapping input examples x in X to class
labels y in Y. The uniform convergence is only ensured when the Shattering
coefficient N(F,2n) has a polynomial growing behavior. This paper proves the
Shattering coefficient for any Hilbert space H containing the input space X and
discusses its effects in terms of learning guarantees for supervised machine
algorithms.
