From linear to metric functional analysis

Significance Metric geometry is a considerable extension of Riemannian geometry that, in recent decades, has proven very useful. A newer direction described in this article can moreover be viewed as an extension of functional analysis. I try to illustrate why these more recent concepts and results will be of importance, potentially generating new ideas in applied mathematics, for example, deep learning or evolution dynamics. An important objective of this note is therefore also to try to attract the attention of scientists in other disciplines. An extension of the contraction mapping principle and a recent fundamental theorem on the behavior of the composition of randomly selected transformation have vast potential for further applications.