  This article studies constructions of reproducing kernel Banach spaces
(RKBSs) which may be viewed as a generalization of reproducing kernel Hilbert
spaces (RKHSs). A key point is to endow Banach spaces with reproducing kernels
such that machine learning in RKBSs can be well-posed and of easy
implementation. First we verify many advanced properties of the general RKBSs
such as density, continuity, separability, implicit representation, imbedding,
compactness, representer theorem for learning methods, oracle inequality, and
universal approximation. Then, we develop a new concept of generalized Mercer
kernels to construct $p$-norm RKBSs for $1\leq p\leq\infty$. The $p$-norm RKBSs
preserve the same simple format as the Mercer representation of RKHSs.
Moreover, the $p$-norm RKBSs are isometrically equivalent to the standard
$p$-norm spaces of countable sequences. Hence, the $p$-norm RKBSs possess more
geometrical structures than RKHSs including sparsity. The generalized Mercer
kernels also cover many well-known kernels, for example, min kernels, Gaussian
kernels, and power series kernels. Finally, we propose to solve the support
vector machines in the $p$-norm RKBSs, which are to minimize the regularized
empirical risks over the $p$-norm RKBSs. We show that the infinite dimensional
support vector machines in the $p$-norm RKBSs can be equivalently transferred
to finite dimensional convex optimization problems such that we obtain the
finite dimensional representations of the support vector machine solutions for
practical applications. In particular, we verify that some special support
vector machines in the $1$-norm RKBSs are equivalent to the classical $1$-norm
sparse regressions. This gives fundamental supports of a novel learning tool
called sparse learning methods to be investigated in our next research project.
