A characterization of inner product spaces via norming vectors
Guillaume Aubrun, Mathis Cavichioli
TL;DR
This work characterizes finite‑dimensional inner product spaces by a property of norming vectors associated with endomorphisms, extending a real‑field result of Sain and Paul to complex scalars. It centers on the set $N(u)=\{ x : \|u x\| = \|u\|_{op}\,|x| \}$ and proves that $X$ is inner product iff $N(u)$ is a linear subspace for all $u$. The authors establish a key proposition: for any norm on $K^n$ and $Q\in GL_n(K)$, the set $N(AQ)$ spans $K^n$ for a suitably chosen $A$ of maximal determinant in $M_n^+(K)\cap C_Q$, using a Hahn–Banach–style duality framework and a perturbation lemma. This yields a self-contained finite‑dimensional proof that extends to complex fields, and shows that the isometry group of $X$ must be the full orthogonal group, forcing the norm to be Euclidean up to a scalar. The result provides a robust, endomorphism‑based criterion for recognizing inner product spaces with potential implications for related geometric characterizations.
Abstract
A finite-dimensional normed space is an inner product space if and only if the set of norming vectors of any endomorphism is a linear subspace. This theorem was proved by Sain and Paul for real scalars. In this paper, we give a different proof which also extends to the case of complex scalars.