Euclidean geometry | Topological spaces | Linear algebra | Norms (mathematics)

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension, including the three-dimensional space and the Euclidean plane (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his Elements, with the great innovation of proving all properties of the space as theorems, by starting from a few fundamental properties, called postulates, which either were considered as evident (for example, there is exactly one straight line passing through two points), or seemed impossible to prove (parallel postulate). After the introduction at the end of 19th century of non-Euclidean geometries, the old postulates were re-formalized to define Euclidean spaces through axiomatic theory. Another definition of Euclidean spaces by means of vector spaces and linear algebra has been shown to be equivalent to the axiomatic definition. It is this definition that is more commonly used in modern mathematics, and detailed in this article. In all definitions, Euclidean spaces consist of points, which are defined only by the properties that they must have for forming a Euclidean space. There is essentially only one Euclidean space of each dimension; that is, all Euclidean spaces of a given dimension are isomorphic. Therefore, in many cases, it is possible to work with a specific Euclidean space, which is generally the real n-space equipped with the dot product. An isomorphism from a Euclidean space to associates with each point an n-tuple of real numbers which locate that point in the Euclidean space and are called the Cartesian coordinates of that point. (Wikipedia).

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From playlist Science Unplugged: Physics

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

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

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From playlist Science Unplugged: Special Relativity

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

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From playlist Linear Algebra

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From playlist Multivariable Spaces and Functions

CS224W: Machine Learning with Graphs | 2021 | Lecture 19.2 - Hyperbolic Graph Embeddings

For more information about Stanford’s Artificial Intelligence professional and graduate programs, visit: https://stanford.io/3Brc7vN Jure Leskovec Computer Science, PhD In previous lectures, we focused on graph representation learning in Euclidean embedding spaces. In this lecture, we in

From playlist Stanford CS224W: Machine Learning with Graphs

Colloquium MathAlp 2019 - Claude Lebrun

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From playlist Colloquiums MathAlp

Johnathan Bush (11/5/21): Maps of Čech and Vietoris–Rips complexes into euclidean spaces

We say a continuous injective map from a topological space to k-dimensional euclidean space is simplex-preserving if the image of each set of at most k+1 distinct points is affinely independent. We will describe how simplex-preserving maps can be useful in the study of Čech and Vietoris–Ri

From playlist Vietoris-Rips Seminar

Today, we begin the manifolds series by introducing the idea of a topological manifold, a special type of topological space which is locally homeomorphic to Euclidean space.

From playlist Manifolds

Stability of the spacetime positive mass theorem in spherical symmetry - Marcus Khuri

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From playlist Mathematics

Entanglement in QFT and Quantum Gravity (Lecture 1) by Tom Hartman

PROGRAM KAVLI ASIAN WINTER SCHOOL (KAWS) ON STRINGS, PARTICLES AND COSMOLOGY (ONLINE) ORGANIZERS Francesco Benini (SISSA, Italy), Bartek Czech (Tsinghua University, China), Dongmin Gang (Seoul National University, South Korea), Sungjay Lee (Korea Institute for Advanced Study, South Korea

From playlist Kavli Asian Winter School (KAWS) on Strings, Particles and Cosmology (ONLINE) - 2022

Selling Real Estate in Hyperbolic Space - Mel Slugbate (Colin Adams) [1996]

slides for this talk: http://www.msri.org/realvideo/ln/msri/1996/conv/adams/1/banner/01.html Conversations between Mathematics Teachers and Mathematics Researchers December 11, 1996 Selling Real Estate in Hyperbolic Space: Investment Opportunities for the 90's" Mel Slugbate (Colin Adams)

From playlist Mathematics

Stefan Wenger - 21 September 2016

Wenger, Stefan "“Plateau’s problem in metric spaces and applications”"

From playlist A Mathematical Tribute to Ennio De Giorgi

This video explains the definition of a vector space and provides examples of vector spaces.

From playlist Vector Spaces

Examples of non-positively curved groups II - Kim Ruane

Women and Mathematics Title: Examples of non-positively curved groups II Speaker: Kim Ruane Affiliation: Tufts University Date: May 24, 2017 For more videos, please visit http://video.ias.edu

From playlist Mathematics