Universal operatorial Noether current associated with a quasi-complete locally convex linear space valued Lagrangian defined on an open set of a product of normed spaces

In the context of locally convex vector valued integrable tensors on open sets of a $n$-dimensional manifold $M$ developped in our previous work \cite{sil09261621}, in this paper we construct a framework in which to address both a variational problem associated with a Lagrangian $\mathcal{L}$, and a conservation law on-shell associated with actions $\phi$ and $\omega$. $\mathcal{L}$ is a $C^p$-map valued in a quasi-complete Hausdorff lcs $\mathcal{Z}$ and defined on an open set $O$ of a normed space, $\phi$ is a $C^p$-action parametrized by an open $A$ of a normed space $T$, acting over the space ${\mathfrak{Z}}^O$ of fields valued in a normed space $Y_0$ and defined on $W$ domain of a chart $\xi$ of $M$; $\omega$ is a $C^p$-action on $W$ parametrized by $A$. Via a result in \cite{sil09261621}, we construct $\mathfrak{A}:{\mathfrak{Z}}^O\times A\to \mathcal{M}(W,\mathcal{Z})$ where $\mathfrak{A}(u,\eta )$ is the action functional associated with $\mathcal{L}$, $u$ and $\eta$. Here $\mathcal{M}(W,\mathcal{Z})={\mathcal{L}}_s(\mathcal{H}(W),\mathcal{Z})$ is the lcs of measures on $W$ and valued in $\mathcal{Z}$. By letting $\mathcal{A}(u):A\to \mathcal{M}(W,\mathcal{Z})$ be such that $\mathcal{A}(u)(\eta )=\mathfrak{A}(u,\eta )$, we state that $\mathcal{A}(u)$ is differentiable off-shell, find its differential map off and on-shell. We establish that if the differential map of $\mathcal{A}(u)$ on $0$ is $0$, then $\mathcal{N}[u]$ is conserved on-shell, and if $p=\mathrm{\infty }$, then the divergence of $\mathcal{N}[u]$ equals $0$, where the divergence of a lcs valued vector field is defined and analyzed in \cite{sil09261621}. Here $\mathcal{N}[u]$ is a ${\mathcal{L}}_b(T,\mathcal{Z})$-valued $C^{(p-1)}$-vector field of $M$ on $U=\xi (W)$.