We present some basic elements of the theory of generalised Br\`{e}gman relative entropies over nonreflexive Banach spaces. Using nonlinear embeddings of Banach spaces together with the Euler--Legendre functions, this approach unifies two former approaches to Br\`{e}gman relative entropy: one based on reflexive Banach spaces, another based on differential geometry. This construction allows to extend Br\`{e}gman relative entropies, and related geometric and operator structures, to arbitrary-dimensional state spaces of probability, quantum, and postquantum theory. We give several examples, not considered previously in the literature.

翻译：本文介绍了非自反巴拿赫空间上广义布雷格曼相对熵理论的基本要素。通过结合巴拿赫空间的非线性嵌入与欧拉-勒让德函数，该方法统一了先前关于布雷格曼相对熵的两种处理方式：一种基于自反巴拿赫空间，另一种基于微分几何。这一构造使得布雷格曼相对熵及其相关的几何与算子结构能够推广到概率论、量子理论及后量子理论的任意维状态空间。本文还给出了多个此前文献中未曾探讨的实例。