  This paper provides some first steps in developing empirical process theory
for functions taking values in a vector space. Our main results provide bounds
on the entropy of classes of smooth functions taking values in a Hilbert space,
by leveraging theory from differential calculus of vector-valued functions and
fractal dimension theory of metric spaces. We demonstrate how these entropy
bounds can be used to show the uniform law of large numbers and asymptotic
equicontinuity of the function classes, and also apply it to statistical
learning theory in which the output space is a Hilbert space. We conclude with
a discussion on the extension of Rademacher complexities to vector-valued
function classes.
