  In this article, we study the whole theory of regularized learning for
linear-functional data in Banach spaces including representer theorems,
pseudo-approximation theorems, and convergence theorems. The input training
data are composed of linear functionals in the predual space of the Banach
space to represent the discrete local information of multimodel data and
multiscale models. The training data and the multi-loss functions are used to
compute the empirical risks to approximate the expected risks, and the
regularized learning is to minimize the regularized empirical risks over the
Banach spaces. The exact solutions of the original problems are approximated
globally by the regularized learning even if the original problems are unknown
or unformulated. In the convergence theorems, we show the convergence of the
approximate solutions to the exact solutions by the weak* topology of the
Banach space. Moreover, the theorems of the regularized learning are applied to
solve many problems of machine learning such as support vector machines and
artificial neural networks.
