Inclination of subspaces and decomposition of electromagnetic fields into potential and vortex components

Abstract

Using the notion of inclination of two subspaces $L$ and $M$ of Hilbert space $\mathcal{H}$, we prove the theorem on the extension of linear continuous functionals defined on the subspace $L$ to $\mathcal{H}$ so that the extended functionals vanish on the subspace $M$. We apply this theorem to study the question of decomposition of the electromagnetic field in resonator with ideally conducting walls into potential and vortex components and derive the Korn-type inequality for vortex fields.