  Recently, there has been emerging interest in constructing reproducing kernel
Banach spaces (RKBS) for applied and theoretical purposes such as machine
learning, sampling reconstruction, sparse approximation and functional
analysis. Existing constructions include the reflexive RKBS via a bilinear
form, the semi-inner-product RKBS, the RKBS with $\ell^1$ norm, the $p$-norm
RKBS via generalized Mercer kernels, etc. The definitions of RKBS and the
associated reproducing kernel in those references are dependent on the
construction. Moreover, relations among those constructions are unclear. We
explore a generic definition of RKBS and the reproducing kernel for RKBS that
is independent of construction. Furthermore, we propose a framework of
constructing RKBSs that unifies existing constructions mentioned above via a
continuous bilinear form and a pair of feature maps. A new class of Orlicz
RKBSs is proposed. Finally, we develop representer theorems for machine
learning in RKBSs constructed in our framework, which also unifies representer
theorems in existing RKBSs.
