  This note is propaedeutic to the forthcoming work \cite{sil}; here we develop
the terminology and results required by that paper. More specifically we
introduce the concept of scalarly essentially integrable locally convex
vector-valued tensor fields on a smooth manifold, generalize on them the usual
operations, in case the manifold is oriented define the weak integral of
scalarly essentially integrable locally convex vector-valued maximal forms and
finally establish the extension of Stokes theorem for smooth locally convex
vector-valued forms. This approach to the basic theory of scalarly essentially
integrable and smooth locally convex vector-valued tensor fields seems to us to
be new. Specifically are new (1) the definition of the space of scalarly
essentially integrable locally convex vector-valued tensor fields as a
$\mathcal{A}(U)$-tensor product, although motivated by a result in the usual
smooth and real-valued context; (2) the procedure of
$\mathcal{A}(U)$-linearizing $\mathcal{A}(U)$-bilinear maps in order to extend
the usual operations especially the wedge product; (3) the exploitation of the
uniqueness decomposition of the $\mathcal{A}(U)$-tensor product with a free
module in order to define not only (a) the exterior differential of smooth
locally convex vector-valued forms, but also (b) the weak integral of scalarly
essentially integrable locally convex vector-valued maximal forms; (4) the use
of the projective topological tensor product theory to define the wedge
product.
