From linear to metric functional analysis.

This article presents the beginning of a metric functional analysis. A major notion is metric functionals which extends that of horofunctions in metric geometry. Applications of the main tools are found in a wide variety of subjects such as random walks on groups, complex dynamics, surface topology, deep learning, evolution equations, and game theory, thus branching well outside of pure mathematics. In several cases, linear notions fail to describe linear phenomena that are naturally captured by metric concepts. An extension of the mean ergodic theorem testifies to this. A general metric fixed-point theorem is also proved.