# The triangle inequality for graded real vector spaces

## TL;DR

This paper proves that a natural homogeneous norm on graded Lie algebras satisfies the triangle inequality, confirming a key property for these mathematical structures and answering a previously posed question.

## Contribution

It establishes the triangle inequality for a natural candidate norm on graded Lie algebras of any length, advancing understanding of their geometric properties.

## Key findings

- The candidate norm satisfies the triangle inequality.
- The result applies to graded Lie algebras of any length.
- Answers Moskowitz's question affirmatively.

## Abstract

In this paper, we prove that a natural candidate for a homogeneous norm on a graded Lie algebra of any length satisfies the triangle inequality which answers Moskowitz's question.