Mathematical structures | Metric spaces | Topology | Uniform spaces | Topological spaces | Mathematical analysis
In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and therefore admit the structure of a metric space, including Riemannian manifolds, normed vector spaces, and graphs. In abstract algebra, the p-adic numbers arise as elements of the completion of a metric structure on the rational numbers. Metric spaces are also studied in their own right in metric geometry and analysis on metric spaces. Many of the basic notions of mathematical analysis, including balls, completeness, as well as uniform, Lipschitz, and Hölder continuity, can be defined in the setting of metric spaces. Other notions, such as continuity, compactness, and open and closed sets, can be defined for metric spaces, but also in the even more general setting of topological spaces. (Wikipedia).
This lecture is on Introduction to Higher Mathematics (Proofs). For more see http://calculus123.com.
From playlist Proofs
Metric space definition and examples. Welcome to the beautiful world of topology and analysis! In this video, I present the important concept of a metric space, and give 10 examples. The idea of a metric space is to generalize the concept of absolute values and distances to sets more gener
From playlist Topology
What is a metric space? An example
This is a basic introduction to the idea of a metric space. I introduce the idea of a metric and a metric space framed within the context of R^n. I show that a particular distance function satisfies the conditions of being a metric.
From playlist Mathematical analysis and applications
Introduction to Metric Spaces - Definition of a Metric. - The metric on R - The Euclidean Metric on R^n - A metric on the set of all bounded functions - The discrete metric
From playlist Topology
Complete metric space: example & proof
This video discusses an example of particular metric space that is complete. The completeness is proved with details provided. Such ideas are seen in branches of analysis.
From playlist Mathematical analysis and applications
MAST30026 Lecture 2: Examples of spaces (Part 1)
I started with the definition of a metric space, we briefly discussed the example of Euclidean space (proofs next time) and then I started to explain a few natural metrics on the circle. Lecture notes: http://therisingsea.org/notes/mast30026/lecture2.pdf The class webpage: http://therisin
From playlist MAST30026 Metric and Hilbert spaces
Weird notions of "distance" || Intro to Metric Spaces
Visit https://brilliant.org/TreforBazett/ to get started learning STEM for free, and the first 200 people will get 20% off their annual premium subscription. Check out my MATH MERCH line in collaboration with Beautiful Equations ►https://www.beautifulequation.com/pages/trefor Weird, fun
From playlist Cool Math Series
This video is about metric spaces and some of their basic properties.
From playlist Basics: Topology
MAST30026 Lecture 1: What is space? (Part 1)
I started with three dictionary definitions of "space" and briefly discussed them, before moving on to a survey of the standard abstract notions of space used in mathematics, including metric, topological and Hilbert spaces. In the remainder of the lecture I discussed the connection betwee
From playlist MAST30026 Metric and Hilbert spaces
Pawel Grzegrzolka - Asymptotic dimension of fuzzy metric spaces
38th Annual Geometric Topology Workshop (Online), June 15-17, 2021 Pawel Grzegrzolka, Stanford University Title: Asymptotic dimension of fuzzy metric spaces Abstract: In this talk, we will discuss asymptotic dimension of fuzzy metric spaces. After a short introduction to fuzzy metric spac
From playlist 38th Annual Geometric Topology Workshop (Online), June 15-17, 2021
Minerva Lectures 2013 - Assaf Naor Talk 2: Dichotomies and universality in metric embeddings
For more information, please see: http://www.math.princeton.edu/events/seminars/minerva-lectures/minerva-lecture-ii-dichotomies-and-universality-metric-embeddings
From playlist Minerva Lectures - Assaf Naor
Woojin Kim (6/15/20): Spatiotemporal persistent homology for dynamic metric spaces
Title: Spatiotemporal persistent homology for dynamic metric spaces Abstract: Characterizing the dynamics of time-evolving data within the framework of topological data analysis (TDA) has been attracting increasingly more attention. Popular instances of time-evolving data include flocking
From playlist ATMCS/AATRN 2020
Nicolò Zava (3/17/23): Every stable invariant of finite metric spaces produces false positives
In computational topology and geometry, the Gromov-Hausdorff distance between metric spaces provides a theoretical framework to tackle the problem of shape recognition and comparison. However, the direct computation of the Gromov-Hausdorff distance between finite metric spaces is known to
From playlist Vietoris-Rips Seminar
Bei Wang (5/7/21): Homotopy Types of Vietoris–Rips Complexes of Metric Gluings
We study Vietoris-Rips complexes of metric wedge sums and metric gluings. We show that the Vietoris-Rips complex of a wedge sum, equipped with a natural metric, is homotopy equivalent to the wedge sum of the Vietoris–Rips complexes. We also provide generalizations for when two metric space
From playlist Vietoris-Rips Seminar
MAST30026 Lecture 13: Metrics on function spaces (Part 2)
I discussed pointwise and uniform convergence of functions, proved that the uniform limit of continuous functions is continuous, and used that to prove that Cts(X,Y) is a complete metric space with respect to the sup metric if X is compact and Y is a complete metric space. Lecture notes:
From playlist MAST30026 Metric and Hilbert spaces
What is General Relativity? Lesson 5: The Catalogue of Spacetimes
What is General Relativity? Lesson 5: The Catalogue of Spacetimes - Minkowski Spacetime I invite you to download the Catalog of Spacetimes at : https://arxiv.org/abs/0904.4184 to use as a reference for the rest of the course.
From playlist What is General Relativity?
Minerva Lectures 2013 - Assaf Naor Talk 1: An introduction to the Ribe program
For more information, please see: http://www.math.princeton.edu/events/seminars/minerva-lectures/minerva-lecture-i-introduction-ribe-program
From playlist Minerva Lectures - Assaf Naor
What is General Relativity? Lesson 8: Intro to the metric connection and the induced metric.
This lesson is an introduction to the concept of the metric connection followed by a long exercise in classical differential geometry. It is a long lesson because I complete a full example: the derivation of the metric of the "glome" induced by the Euclidean metric of 4-dimensional space.
From playlist What is General Relativity?
What is General Relativity? Lesson 21: Geodesic Equation Part 4: Null and Conformal Geodesics
This video is about What is General Relativity? Lesson 21: Null and Conformal Geodesics In this lecture we clean up a few topics: 1) The Lagrangian for null geodesics and, 2) The relationship between geodesics of conformally related metrics. Check out the forums at http://xylyxylyx.freef
From playlist What is General Relativity?
Metric Units of Measurement (1 of 3: Overview of various metric units)
More resources available at www.misterwootube.com
From playlist Applications of Measurement