Non-normability of spaces of Keplerian orbits

TL;DR

This paper demonstrates that certain spaces of Keplerian orbits cannot be equipped with a norm compatible with their natural topology or metrics, highlighting fundamental limitations in their mathematical structure.

Contribution

It establishes the non-normability of various spaces of Keplerian orbits, providing new insights into their topological and metric properties.

Findings

Spaces of Keplerian orbits are non-normable.

Non-normability holds for all orbits and elliptic orbits with marked pericenter.

Elliptic orbits without marked pericenter also cannot carry a compatible norm.

Abstract

We prove that spaces of Keplerian curvilinear orbits, all orbits and elliptic orbits with marked pericenter cannot carry a norm, compatible with their standard topology. We also prove that the space of Keplerian elliptic orbits without marked pericenter cannot carry a norm, compatible with the natural metrics on it.