Reflexivity and James’ Theorem via Vector-valued Banach Limits

We present a new, concise proof of the result that a Banach space X X is reflexive if it admits a Banach limit on bounded X X -valued sequences such that the limit vector of any input sequence lies in the closed linear span of that sequence. Building on this result, and aiming to demonstrate the utility of vector-valued Banach limits with the linear span property just described, we provide a new proof of James’ theorem characterising reflexive Banach spaces with a Schauder basis. This proof consists of two parts, both employing vector-valued Banach limits, with one part specifically relying on Banach limits that possess the linear span property.