Fekete lemma in Banach spaces

The classical Fekete lemma says that if the sequence of real numbers 𝑎_𝑛 satisfies the inequality 𝑎_𝑛+𝑚 ≤ 𝑎_𝑛 + 𝑎_𝑚for all 𝑛, 𝑚 ∈ ℕ then the limit lim_𝑛→∞ 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|| ≤ ||𝑎_𝑛 +𝑎_𝑚|| for all 𝑛, 𝑚 ∈ ℕ. Then the limit lim_𝑛→∞ 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.

If time permits we will also discuss some generalizations of the Fekete's lemma which restrict the set of pairs (𝑛, 𝑚) for which we know the inequality. Curiously, some of them are no longer true already for 2-dimensional Hilbert spaces 𝑋.

The talk is based on a joint work with Feng Shao.