Hyperbolic Banach spaces

The standard theory of Banach spaces is built upon the notions of vector space, triangle inequality and Cauchy completeness. Here we propose a hyperbolic' variant of this elliptic’ framework where general linear combinations are replaced by linear combinations with non-negative coefficients, triangle inequality is replaced by reverse triangle inequality and Cauchy completeness is replaced by the order-theoretic notion of directed completeness. The motivation for our investigation is in non-smooth Lorentzian geometry: we believe that to unlock the full potential of the field, and ultimately extract more informations about the smooth world, some version of Lorentzian functional analysis' is needed, especially in relation to timelike lower Ricci curvature bounds. An example of structure we investigate is obtained by starting with a Banach space, multiplying it by $\mathbb R$ and considering the future cone’ in there. Because of this, some of the results in this manuscript might be read through the lenses of standard Banach spaces theory. From this perspective, the classical Hahn-Banach and Baire category theorems can be seen as consequences of statements obtained here. A different kind of example is that of $L^p$ spaces for $p\leq1$. Their structure and natural duality relations fit particularly well in our framework, to the extent that they have been an important source of inspiration for the axiomatization chosen in this paper. We also investigate the notion of directed completeness regardless of any algebraic structure, as we believe it is central even in the finite-dimensional non-smooth Lorentzian framework, for instance to achieve a compactness theorem à la Gromov. This study unveils connections between Geroch-Kronheimer-Penrose’s concept of ideal point in a spacetime, Beppo Levi’s monotone convergence theorem and certain aspects of domain theory.

💡 Research Summary

This paper by Nicola Gigli lays the foundational groundwork for a new branch of functional analysis tailored to Lorentzian signature geometry, analogous to how classical Banach space theory underpins analysis in Riemannian settings. The core innovation is a systematic “hyperbolic” reformulation of the three pillars of standard “elliptic” Banach space theory: the vector space structure, the triangle inequality, and Cauchy completeness.

The proposed framework replaces general linear combinations with combinations using only non-negative coefficients, leading to structures called “wedges.” The triangle inequality is substituted with the reverse triangle inequality, intrinsic to Lorentzian distance functions. Most significantly, Cauchy completeness is replaced by the order-theoretic concept of “directed completeness,” where every directed subset has a supremum. This choice is motivated by its natural fit with Lorentzian causality, its well-behaved completion procedure (which resembles adding future infinity to a spacetime), and its connection to the Monotone Convergence Property evident in key examples like L^p spaces for p ≤ 1.

The paper is structured in three main parts. First, it develops the pure order theory of directed completeness and presents a universal construction for the “directed completion” of a partial order. This completion is linked to the Geroch-Kronheimer-Penrose construction of ideal points in general relativity.

Second, the algebraic structure is integrated. A “cone” is defined as a directed complete wedge. The natural preorder (v ≤ w iff ∃z: v+z=w) becomes a partial order in “meaningful wedges.” The study focuses on “cones with joins,” which are lattice-ordered and allow for the definition of a crucial concept: the “part at infinity” εv of an element. This concept enables a weakened cancellation law, which is instrumental in proving the paper’s central extension theorem (Theorem 3.17). This theorem provides conditions under which a linear map defined on a sub-wedge and bounded between a subadditive and a superadditive functional can be extended to the entire cone, serving as a hyperbolic analogue of the Hahn-Banach theorem.

Third, the theory is validated through examples and applications. The L^p spaces for p ≤ 1, including p=0 and negative exponents, are shown to fit perfectly within this framework, and their natural duality for conjugate exponents (1/p + 1/q = 1) was a key inspiration for the axiomatization. “Banach spacetimes,” built from the future cone in the product of a Banach space with R, demonstrate how classical results like the Hahn-Banach and Baire category theorems can be recovered as special cases. Further examples explore duality between measures and functions, matrix spaces, the Brunn-Minkowski inequality reinterpreted as a reverse triangle inequality, and finite-dimensional cases.

In summary, this work proposes a coherent and novel functional-analytic framework based on cones, reverse triangle inequality, and directed completeness. It aims to provide the necessary analytic tools for developing a robust calculus in non-smooth Lorentzian geometry, particularly for theories involving timelike lower Ricci curvature bounds. While the theory is in its early stages, it successfully establishes core definitions, key theorems, and convincing examples, opening a promising avenue for future research at the intersection of functional analysis, order theory, and Lorentzian geometry.