Analysis Of Metric Spaces
Analysis Of Metric Spaces
Published 5/2023
MP4 | Video: h264, 1280x720 | Audio: AAC, 44.1 KHz
Language: English | Size: 11.60 GB | Duration: 28h 33m
The world according to Rudin
What you'll learn
Effectively locate and use the information needed to prove theorems and establish mathematical results.
Effectively write mathematical solutions in a clear and concise manner.
Demonstrate an intuitive understanding of set theory and metric spaces
Understand the classic book Principles of Mathematical Analysis by Rudin
Requirements
Mathematical maturity. In other words, be interested in mathematics.
Description
The methods of calculus are limited to Euclidean spaces. In this course, we show how the incredibly powerful tools of calculus, beginning with the limit concept, can be generalized to so-called metric spaces. Almost every space used in advanced analysis is in fact a metric space, and limits in metric spaces are a universal language for advanced analysis. The basic techniques of calculus were invented for the real line R. What should we do when we want to handle something more general than R? The fundamental notions of calculus begin with the idea that one point of R can be close to another point of R, and this is called "Approximation" or "taking a limit." Metrics are a way to transfer this key notion of being "close to" to a more general setting. The "points" of a metric space can be complicated objects in their own right. For example, they may themselves be functions on some other space. Ideas like this are ubiquitous in advanced mathematics today. One tries to throw away complicated details of the space being considered, and this makes it easier to see which theorem or technique can be applied next. In this way, the mathematician tries to avoid getting overwhelmed by the details, or to say it differently, we try to see the overall forest rather than the trees.
Overview
Section 1: Introduction
Lecture 1 Introduction and Chapter 2 of Rudin
Lecture 2 Basic Set Theory
Lecture 3 Sequences
Lecture 4 Real and Rational Numbers
Lecture 5 The real numbers are uncountable
Lecture 6 Russel's paradox: not _everything_ can be a set
Lecture 7 More on Paradox... and the start of Metric Spaces
Lecture 8 Neighbourhoods in Metric space
Lecture 9 Interior Points, Open, Closed, and Clopen
Lecture 10 Dense Sets and Perfect sets
Lecture 11 Closed, bounded, compact
Lecture 12 Open is not the opposite of closed
Lecture 13 Not limits, but Limit Points. Closures.
Lecture 14 Compactness: a very clever generalization of closed intervals of the number line
Lecture 15 More on compactness
Lecture 16 Review
Lecture 17 Properties of compact sets and of perfect sets
Lecture 18 The real line: Sequences and Series
Lecture 19 The Cantor set: Is it connected?
Lecture 20 Sequences in Metric Spaces
Lecture 21 More on sequences in Metric Spaces
Lecture 22 Another clever generalization: completeness through Cauchy sequences
Lecture 23 Comparing the two kinds of completeness
Lecture 24 Continuity: a property of maps of metric spaces
Have you tried to read the classic book Introduction to Mathematical Analysis by Rudin and been stimied? Take this course!
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