Operations on $\mathcal{M}{\otimes }_{\mathcal{A}(U)}{\mathrm{Alt}}^{\bullet }(U,M)$. Stokes theorem for locally convex linear space valued forms. Divergence theorem for locally convex linear space valued vector fields

Abstract

We extend to $\mathcal{M}{\otimes }_{\mathcal{A}(U)}{\mathfrak{T}}_{\bullet }^{\bullet }(U,M)$ the usual operations defined on ${\mathfrak{T}}_{\bullet }^{\bullet }(U,M)$, then by employing the properties of the projective tensor product of locally convex spaces, we generalize to $\mathcal{M}{\otimes }_{\mathcal{A}(U)}{\mathrm{Alt}}^{\bullet }(U,M)$ the usual wedge product, insertion operator and exterior differential defined on ${\mathrm{Alt}}^{\bullet }(U,M)$ and establish their properties. Here $M$ is a smooth finite dimensional manifold, $U$ an open submanifold of $M$, $\mathcal{A}(U)$ the ring of smooth maps on $U$, ${\mathfrak{T}}_{\bullet }^{\bullet }(U,M)$ the $\mathcal{A}(U)$-module of smooth tensor fields of $M$ defined on $U$, ${\mathrm{Alt}}^{\bullet }(U,M)$ the $\mathcal{A}(U)$-module of smooth alternating tensor fields of $M$ defined on $U$, while $\mathcal{M}$ is a $\mathcal{A}(U)$-module. $\mathcal{M}$ depending by the operation might be either general, or one of the following function spaces: ${\mathcal{L}}_c^1(U,G,\lambda )$, ${\mathcal{L}}_{loc}^1(U,G,\lambda )$, ${\mathcal{B}}^p(U,G)$ with $G$ a Hausdorff locally convex space. This framenwork permits to construct the divergence of a $G$-valued $p$-times continuously differentiable vector field, to define the weak integral of $G$-valued compactly supported scalarly integrable maximal forms, to associate a $G$-valued measure with any locally integrable $G$-valued form, to establish a Stokes type theorem for $G$-valued compactly supported $p$-times continuously differentiable forms and to obtain a divergence type theorem for $G$-valued $p$-times continously differentiable vector fields.

arXiv commentDefinition generalized and new results established