Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Norms and Distances in Analysis II

In course

MATH-106(f): Analysis II

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles de plusieurs variables. ...

Description

This lecture covers fundamental concepts in mathematical analysis, focusing on norms and distances. The instructor begins by introducing the definitions and properties of norms, emphasizing positivity, homogeneity, and the triangle inequality. The discussion progresses to distances, defined as the norm of the difference between two points, and explores their properties, including symmetry and the triangle inequality. The lecture also delves into open and closed sets, explaining how to determine whether a set is open or closed based on its boundary. The instructor illustrates these concepts with examples, demonstrating how to visualize sets in a two-dimensional space. The importance of understanding the boundary of a set is highlighted, as it plays a crucial role in classifying sets as open or closed. The lecture concludes with a discussion on the interior and closure of sets, providing a comprehensive overview of these essential topics in analysis.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructors (3)

Adélie Eliane Garin

Boris Buffoni

David Valentin Strütt

Official source

Ontological neighbourhood

Mathematics

Topology: General topology

Related concepts (32)

Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. This should not be confused with a closed manifold. By definition, a subset of a topological space is called if its complement is an open subset of ; that is, if A set is closed in if and only if it is equal to its closure in Equivalently, a set is closed if and only if it contains all of its limit points.

Clopen set

In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical definitions are not mutually exclusive. A set is closed if its complement is open, which leaves the possibility of an open set whose complement is also open, making both sets both open closed, and therefore clopen.

Open and closed maps

In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function is open if for any open set in the is open in Likewise, a closed map is a function that maps closed sets to closed sets. A map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa. Open and closed maps are not necessarily continuous.

Norm (mathematics)

In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance in a Euclidean space is defined by a norm on the associated Euclidean vector space, called the Euclidean norm, the 2-norm, or, sometimes, the magnitude of the vector.

Topology

In mathematics, topology (from the Greek words τόπος, and λόγος) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity.

Related lectures (51)

Functions: Definitions and Notations

Covers the generalities of functions, including the definition of an application between sets and the uniqueness of elements in the image set.

Vector Spaces and Topology

Covers normed vector spaces, topology in R^n, and the principle of drawers as a demonstration method.

Linear Algebra: Applications and Definitions

Covers the definition of applications, image of elements, and direct and reciprocal images.

Geometric Considerations in Rn

Covers the concept of intervals in Rn using geometric balls and defines open and closed sets, interior points, boundaries, closures, bounded domains, and compact sets.

Quotient Topology