Lecture

Demonstration of Theorem on Compact Functions

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

In course

MATH-106(e): Analysis II

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

Description

This lecture covers a detailed demonstration of a theorem related to continuous functions on a compact set, focusing on the proof that a function attains its minimum and maximum. The instructor explains the concept of boundedness on a compact set, the distinction between sequences of function values and points, and the convergence of subsequences. The lecture also delves into the properties of supremum and infimum values, using the definitions to show convergence. Additionally, an example of a non-regular boundary curve is presented, illustrating a construction method that results in a continuous curve without self-intersections. The lecture concludes with insights on solving systems of equations, analyzing limits, and determining open and closed sets in relation to compactness.

Instructor

Anna Lachowska

Teaching & PhD Courses Algebra MATH-310 This is an introduction to modern algebra: groups, rings and fields. Analysis I MATH-101(e) We study the fundamental concepts of analysis, calculus and the integral of real-valued functions of a real variable. Analysis II MATH-106(e) The course studies fundamental concepts of analysis and the calculus of functions of several variables. Representation theory I - finite groups MATH-314 This is a standard c

Official source

Ontological neighbourhood

Mathematics

Topology: General topology

Related concepts (32)

Compact convergence

In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence that generalizes the idea of uniform convergence. It is associated with the compact-open topology. Let be a topological space and be a metric space. A sequence of functions is said to converge compactly as to some function if, for every compact set , uniformly on as . This means that for all compact , If and with their usual topologies, with , then converges compactly to the constant function with value 0, but not uniformly.

Limit (mathematics)

In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals. The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to and direct limit in . In formulas, a limit of a function is usually written as (although a few authors use "Lt" instead of "lim") and is read as "the limit of f of x as x approaches c equals L".

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.

Compact space

In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all limiting values of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval [0,1] would be compact.

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.

Related lectures (62)

Properties of Convergence: Sequences and Topology

Discusses the properties of sequences, convergence, and their relationship with topology and compactness.

Uniqueness of Solutions: Cauchy-Lipschitz Theorem

Covers the uniqueness of solutions in differential equations, focusing on the Cauchy-Lipschitz theorem and its implications for local and global solutions.

Convergence and Compactness in R^n

Explores adhesion, convergence, closed sets, compact subsets, and examples of subsets in R^n.

Continuous functions on a closed bounded interval

Covers limits, range of continuous functions, and uniform continuity on closed intervals.

Limit of Functions: Convergence and Boundedness