Norms as a function of p are linearly independent in finite dimensions

TL;DR

This paper proves that in finite-dimensional spaces, p-norms are linearly independent functions of p, meaning no universal linear relations exist among them across all p values.

Contribution

It establishes the linear independence of p-norms in finite dimensions using complex analytic continuation, a novel theoretical result.

Findings

No non-trivial linear dependencies among p-norms for all p in finite dimensions

p-norms are linearly independent functions of p

Proof employs complex analytic continuation

Abstract

We show that there are no non-trivial linear dependencies among p-norms of vectors in finite dimensions that hold for all p. The proof is by complex analytic continuation.