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Banach空间上广义Brègman相对熵理论的基本要素。通过利用Banach空间的非线性嵌入以及Euler-Legendre函数，该方法统一了先前两种关于Brègman相对熵的方法：一种基于自反Banach空间，另一种基于微分几何。这一构造使得Brègman相对熵及其相关的几何与算子结构能够推广至概率论、量子理论及后量子理论中任意维度的状态空间。本文给出了多个此前文献中未曾涉及的例子。