Vector-valued Banach limits and the linear span property

This paper explores the connections between vector-valued Banach limits and weak compactness in Banach spaces. We show that a Banach space, \({X}\), is reflexive if it admits a Banach limit on bounded \({X}\)-valued sequences such that, for any input sequence, the corresponding limit vector lies in the closed linear span of that sequence. This conclusion is based on proving that the existence of a vector-valued Banach limit with the aforementioned linear span property implies the weak compactness of the closed unit ball of the underlying Banach space. Furthermore, we extend the above result by establishing a characterisation of the relative weak compactness of bounded sets in Banach spaces. The characterisation states that a bounded set is relatively weakly compact if, for every sequence in the set, there exists a vector-valued Banach limit on the smallest shift-invariant linear space containing the sequence and all vector-valued constant sequences, such that, for any input sequence, the corresponding limit vector lies in the closed linear span of that sequence.