Fekete's lemma in Banach spaces

TL;DR

This paper extends Fekete's lemma to sequences of vectors in uniformly convex Banach spaces, establishing the existence of a limit for normalized sequences under a subadditivity condition.

Contribution

It generalizes Fekete's subadditivity lemma from real sequences to Banach space-valued sequences with a specific norm inequality.

Findings

Proves the existence of the limit of v_n/n in uniformly convex Banach spaces.

Extends classical Fekete's lemma to vector-valued sequences.

Provides a new tool for analyzing subadditive sequences in Banach spaces.

Abstract

For a sequence of vectors $\{v_n\}_{n\in\mathbb{N}}$ in the uniformly convex Banach space $X$ which for all $n, m\in \mathbb{N}$ satisfy $\|v_{n+m}\|\le \|v_n + v_m\|$ we show the existence of the limit $\lim_{n\to \infty} \frac{v_n}{n}$. This extends the classical Fekete's subadditivite lemma to Banach space-valued sequences.