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| Metrics (mathematics) | |
|---|---|
| Name | Metric (mathematics) |
| Field | Topology; Analysis; Geometry |
| Introduced | 19th century |
| Notable | Maurice Fréchet; Felix Hausdorff; Maurice Fréchet; Karl Menger |
Metrics (mathematics) are functions that assign distances between pairs of points in a set, formalizing notions of proximity used throughout Mathematics and influencing work in Picard's era, Maurice Fréchet, Felix Hausdorff, David Hilbert, and Leonhard Euler. Metrics underpin structures studied by researchers associated with Princeton University, University of Göttingen, École Normale Supérieure, Cambridge University, and ETH Zurich and serve as foundational tools in domains connected to John von Neumann, Andrey Kolmogorov, Norbert Wiener, Israel Gelfand, and Kurt Gödel.
A metric on a set X is a function d: X × X → ℝ satisfying positivity, symmetry, and the triangle inequality; historical formalizations appear in work of Maurice Fréchet, Felix Hausdorff, Karl Menger, Emmy Noether, and David Hilbert. The axioms guarantee that d(x,y)=0 iff x=y, d(x,y)=d(y,x), and d(x,z) ≤ d(x,y)+d(y,z), a schema used in textbooks influenced by Stefan Banach, Andrey Kolmogorov, Paul Lévy, Émile Borel, and Henri Lebesgue. Metrics generate metric topologies studied by scholars at University of Cambridge and Harvard University and constrain structures in investigations led by Alexander Grothendieck, Jean-Pierre Serre, Michael Atiyah, and Isadore Singer.
Standard examples include the Euclidean metric on ℝ^n, arising in work by René Descartes, Isaac Newton, Carl Friedrich Gauss, Bernhard Riemann, and Srinivasa Ramanujan; the discrete metric related to combinatorial problems studied by Paul Erdős, Elias M. Stein, and Kurt Mahler; and the taxicab (ℓ1) and maximum (ℓ∞) metrics explored by Hermann Minkowski, Georg Cantor, John von Neumann, and Norbert Wiener. Function-space metrics such as the uniform metric and L^p metrics link to Henri Lebesgue, Stefan Banach, Andrey Kolmogorov, Norbert Wiener, and Paul Dirac. Graph metrics emerge in graph theory advanced by Leonhard Euler, Arthur Cayley, Doron Zeilberger, Erdős, and Richard Karp, while Riemannian metrics extend into differential geometry developed by Bernhard Riemann, Élie Cartan, Marcel Berger, Shing-Tung Yau, and William Thurston.
A metric induces a topology with notions of open balls, convergence, and continuity explored in contexts by Felix Hausdorff, Maurice Fréchet, Kazimierz Kuratowski, Jean-Louis Koszul, and John Milnor. Metric properties determine separation axioms tied to work by André Weil, Élie Cartan, Paul Cohen, Henri Poincaré, and René Thom, and influence compactness, connectedness, and completeness considered by Stefan Banach, Andrey Kolmogorov, David Hilbert, Alfred Tarski, and Alexander Grothendieck. Geometric consequences include curvature and geodesic structure central to research by Bernhard Riemann, Élie Cartan, Marcel Berger, Shing-Tung Yau, and Grigori Perelman.
Convergence and Cauchy sequence concepts in metric spaces trace to Augustin-Louis Cauchy, Karl Weierstrass, Stefan Banach, Maurice Fréchet, and David Hilbert. Completeness, as in Banach spaces, was formalized by Stefan Banach and extended by Andrey Kolmogorov, Israel Gelfand, John von Neumann, and Paul Halmos; completeness characterizes limits, fixed-point theorems like those of Stefan Banach and Brouwer, and functional analysis frameworks used by Laurent Schwartz and Alexander Grothendieck.
Techniques for constructing metrics include induced metrics from embeddings (used by John Nash, Henri Cartan, Bernhard Riemann), quotient metrics studied by Jean-Pierre Serre, André Weil, and Alexander Grothendieck, and product metrics reflecting work at Princeton University and University of Bonn. Transformations like conformal changes, rescalings, and Gromov–Hausdorff limits connect to research by Mikhail Gromov, Gromov's collaborators, Perelman, William Thurston, and Grigori Perelman; ultrametrics arise in p-adic analysis tied to Kurt Hensel, Andrey Kolmogorov, and Alexander Grothendieck.
Metric spaces underpin classical analysis and topology in contributions from Stefan Banach, Andrey Kolmogorov, Felix Hausdorff, Maurice Fréchet, and Paul Lévy. They provide settings for operator theory developed by John von Neumann, Israel Gelfand, and Marshall Stone; for probability theory as in work of Andrey Kolmogorov, Paul Lévy, and Émile Borel; and for geometric group theory pioneered by Mikhail Gromov, Gromov's school, William Thurston, and Miroslav Katětov. Metric invariants such as Hausdorff dimension and packing dimension were introduced by Felix Hausdorff, Paul Lévy, Young, and extended by Kenneth Falconer and Benoit Mandelbrot.
Applications span numerical analysis and approximation theory influenced by John von Neumann, Alan Turing, Norbert Wiener, Stefan Banach, and Erdős; machine learning and data science fields associated with Geoffrey Hinton, Yann LeCun, Andrew Ng, Fei-Fei Li, and Michael Jordan; and theoretical physics areas including general relativity tied to Albert Einstein, Bernhard Riemann, Élie Cartan, Roger Penrose, and Stephen Hawking. Extensions include pseudometrics, quasi-metrics, and metric measure spaces developed by Gromov, Grigori Perelman, Terence Tao, Jean Bourgain, and Cedric Villani, with ongoing research at institutions like Massachusetts Institute of Technology, Stanford University, University of Oxford, École Polytechnique, and Institut des Hautes Études Scientifiques.
Category:Topology Category:Metric geometry