A mathematical framework is developed for the analysis of causal fermion systems in the infinite-dimensional setting. It is shown that the regular spacetime point operators form a Banach manifold endowed with a canonical Fréchet-smooth Riemannian metric. The so-called expedient differential calculus is introduced with the purpose of treating derivatives of functions on Banach spaces which are differentiable only in certain directions. A chain rule is proven for Hölder continuous functions which are differentiable on expedient subspaces. These results are made applicable to causal fermion systems by proving that the causal Lagrangian is Hölder continuous. Moreover, Hölder continuity is analyzed for the integrated causal Lagrangian.

开发了一个数学框架，用于分析无限维环境中的因果费米子系统。结果表明，规则时空点算子形成了一个具有规范 Fréchet 平滑黎曼度量的巴拿赫流形。引入所谓的权宜微分是为了处理仅在某些方向上可微的 Banach 空间上的函数的导数。证明了 Hölder 连续函数的链式法则，该函数在权宜子空间上可微。通过证明因果拉格朗日量是 Hölder 连续的，这些结果适用于因果费米子系统。此外，还分析了综合因果拉格朗日函数的 Hölder 连续性。