on relative reproducing kernel banach spaces: definitions, semi-inner product and feature maps

In this paper, a special class of relative reproducing kernel banach spaces a semi-inner product is studied. we extend the concept of relative reproducing kernel hilbert spaces to banach spaces. we present these relative reproducing kernel banach spaces  in terms of the feature maps and establish the separability of the domains when they are  separable. in addition, we prove some theorems concerning feature maps and reproducing kernel banach spaces. and finally, the relative kernels are compared with the  semi-inner ones.