Scalarly Essentially Integrable Locally Convex Vector Valued Tensor Fields. Stokes Theorem

TL;DR

This paper develops a foundational theory for scalarly essentially integrable and smooth locally convex vector-valued tensor fields on manifolds, extending operations and establishing a version of Stokes theorem.

Contribution

It introduces a novel framework for locally convex vector-valued tensor fields using tensor products and extends classical operations and Stokes theorem within this setting.

Findings

Defined the space of tensor fields as a tensor product.

Extended wedge product and exterior differential to vector-valued forms.

Proved Stokes theorem for smooth locally convex vector-valued forms.

Abstract

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.