  Parametric models in vector spaces are shown to possess an associated linear
map. This linear operator leads directly to reproducing kernel Hilbert spaces
and affine- / linear- representations in terms of tensor products. From the
associated linear map analogues of covariance or rather correlation operators
can be formed. The associated linear map in fact provides a factorisation of
the correlation. Its spectral decomposition, and the associated
Karhunen-Lo\`eve- or proper orthogonal decomposition in a tensor product follow
directly. It is shown that all factorisations of a certain class are unitarily
equivalent, as well as that every factorisation induces a different
representation, and vice versa. A completely equivalent spectral and
factorisation analysis can be carried out in kernel space. The relevance of
these abstract constructions is shown on a number of mostly familiar examples,
thus unifying many such constructions under one theoretical umbrella. From the
factorisation one obtains tensor representations, which may be cascaded,
leading to tensors of higher degree. When carried over to a discretised level
in the form of a model order reduction, such factorisations allow very sparse
low-rank approximations which lead to very efficient computations especially in
high dimensions.
