Fekete's lemma 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 $\Vert v_{n+m}\Vert \le \Vert v_n+v_m\Vert$ 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.

12 pages