  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
Fr\'echet-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 H\"older continuous functions which are differentiable on
expedient subspaces. These results are made applicable to causal fermion
systems by proving that the causal Lagrangian is H\"older continuous. Moreover,
H\"older continuity is analyzed for the integrated causal Lagrangian.
