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From linear to metric functional analysis.

Anders Karlsson1,2

  • 1Section de Mathématiques, Université de Genève, 1211 Geneva 4, Switzerland; anders.karlsson@unige.ch.

Proceedings of the National Academy of Sciences of the United States of America
|July 10, 2021
PubMed
Summary
This summary is machine-generated.

This study introduces metric functionals, a new concept extending horofunctions. These tools offer novel ways to analyze complex systems in diverse fields like deep learning and game theory.

Keywords:
ergodic theoremsfixed-point theoremsmetric geometry

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Area of Science:

  • Mathematics
  • Geometry
  • Analysis

Background:

  • Traditional mathematical analysis often relies on linear concepts.
  • Certain phenomena in geometry and dynamics are not adequately described by linear notions.

Purpose of the Study:

  • Introduce metric functional analysis as a new framework.
  • Extend the concept of horofunctions to metric functionals.
  • Demonstrate the broad applicability of these new mathematical tools.

Main Methods:

  • Development of the theory of metric functionals.
  • Application of metric functional concepts to various mathematical and scientific domains.
  • Extension of existing theorems, such as the mean ergodic theorem.

Main Results:

  • Established metric functionals as a powerful extension of horofunctions.
  • Demonstrated successful applications in random walks, complex dynamics, topology, deep learning, evolution equations, and game theory.
  • Proved an extension of the mean ergodic theorem and a general metric fixed-point theorem.

Conclusions:

  • Metric functional analysis provides a robust framework for phenomena not captured by linear analysis.
  • The introduced metric functionals offer significant potential for advancements across multiple scientific disciplines.
  • This work lays the foundation for a new branch of mathematical analysis with wide-ranging implications.