Hyperbolic Banach spaces

Abstract

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 \`a 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.