Speaker
Description
The classical Fekete lemma says that if the sequence of real numbers $a_n$ satisfies the inequality $a_{n+m}\le a_n + a_m$ for all $n, m\in \mathbb{N}$ then the limit $\lim_{n\to \infty} \frac{a_n}{n}$ exists. In this talk we will discuss what happens when $a_n$ are the elements of some Banach space. The main result that we will discuss is the following theorem.
$Theorem$. Let $X$ be a uniformly convex Banach space and let $a_n$ be a sequence of vectors in $X$ such that $||a_{n+m}|| \le ||a_n+a_m||$ for all $n, m\in \mathbb{N}$. Then the limit $\lim_{n\to \infty} \frac{a_n}{n}$ exists.
Interestingly, the condition of uniform convexity is essential -- if $X$ is not convex (that is, if the unit sphere of $X$ contains an interval) then it is not hard to see that the Fekete lemma fails, but even for convex, but not uniformly convex spaces there might be a counterexample.
The talk is based on a joint work with Feng Shao.
