  In this paper we introduce a generalized Sobolev space by defining a
semi-inner product formulated in terms of a vector distributional operator
$\mathbf{P}$ consisting of finitely or countably many distributional operators
$P_n$, which are defined on the dual space of the Schwartz space. The types of
operators we consider include not only differential operators, but also more
general distributional operators such as pseudo-differential operators. We
deduce that a certain appropriate full-space Green function $G$ with respect to
$L:=\mathbf{P}^{\ast T}\mathbf{P}$ now becomes a conditionally positive
definite function. In order to support this claim we ensure that the
distributional adjoint operator $\mathbf{P}^{\ast}$ of $\mathbf{P}$ is
well-defined in the distributional sense. Under sufficient conditions, the
native space (reproducing-kernel Hilbert space) associated with the Green
function $G$ can be isometrically embedded into or even be isometrically
equivalent to a generalized Sobolev space. As an application, we take linear
combinations of translates of the Green function with possibly added polynomial
terms and construct a multivariate minimum-norm interpolant $s_{f,X}$ to data
values sampled from an unknown generalized Sobolev function $f$ at data sites
located in some set $X \subset \mathbb{R}^d$. We provide several examples, such
as Mat\'ern kernels or Gaussian kernels, that illustrate how many
reproducing-kernel Hilbert spaces of well-known reproducing kernels are
isometrically equivalent to a generalized Sobolev space. These examples further
illustrate how we can rescale the Sobolev spaces by the vector distributional
operator $\mathbf{P}$. Introducing the notion of scale as part of the
definition of a generalized Sobolev space may help us to choose the "best"
kernel function for kernel-based approximation methods.
