Stokes-type Integral Equalities for Scalarly Essentially Integrable Locally Convex Vector Valued Forms which are Functions of an Unbounded Spectral Operator

TL;DR

This paper extends Stokes-type integral equalities to scalar integrable forms valued in locally convex spaces, involving unbounded spectral operators, without requiring smoothness or differentiability.

Contribution

It generalizes previous Newton-Leibnitz-type equalities to include unbounded spectral operators and non-smooth forms in locally convex vector spaces.

Findings

Established a Stokes-type integral equality for scalar integrable forms in locally convex spaces.

Extended the Newton-Leibnitz-type equality to unbounded spectral operators.

Forms involved need not be smooth or differentiable.

Abstract

In this work we establish a Stokes-type integral equality for scalarly essentially integrable forms on an orientable smooth manifold with values in the locally convex linear space $\langle B(G),\sigma(B(G),\mathcal{N})\rangle$, where $G$ is a complex Banach space and $\mathcal{N}$ is a suitable linear subspace of the norm dual of $B(G)$. This result widely extends the Newton-Leibnitz-type equality stated in one of our previous articles. To obtain our equality we generalize the main result of that article, and employ the Stokes theorem for smooth locally convex vector valued forms established in a prodromic paper. Two facts are remarkable. Firstly the forms integrated involved in the equality are functions of a possibly unbounded scalar type spectral operator in $G$. Secondly these forms need not be smooth nor even continuously differentiable.