Lecture 6: The Double Dual and the Outer Measure of a Subset of Real Numbers

preview_player
Показать описание
MIT 18.102 Introduction to Functional Analysis, Spring 2021
Instructor: Dr. Casey Rodriguez

We wrap up our discussion of the Hahn-Banach theorem and move onto the basic notions of measure theory: the second major unit of this course. We define the outer measure (a first guess of “measure”) and prove some basic properties of this new tool.

License: Creative Commons BY-NC-SA

  1. MIT OpenCourseWare
  2. Double dual
  3. Lebesgue integration
  4. Lebesgue measure
  5. isometric
  6. outer measure.
Рекомендации по теме
Lecture 6: The 1:20:58

Lecture 6: The Double Dual and the Outer Measure of a Subset of Real Numbers

SVM Dual : 0:15:32

SVM Dual : Data Science Concepts

Dual Programming Part 0:12:25

Dual Programming Part 1: Relationship between the Primal and Dual LP's

Sec 3 6 0:14:20

Sec 3 6 double dual theorem 19

Lecture 8P1: IE 0:43:42

Lecture 8P1: IE 3340 Operations Research - Dual Theory

Lecture 4, Video 0:17:11

Lecture 4, Video 5: The dual view of RS codes

Dot products and 0:14:12

Dot products and duality | Chapter 9, Essence of linear algebra

Doubles Rules | 0:01:21

Doubles Rules | Badminton

How does a 0:00:58

How does a spectrophotometer work?

Dual Space 0:11:24

Dual Space

Doctorate program: Functional 0:32:20

Doctorate program: Functional Analysis - Lecture 14C: A dual variational problem in optimal control

Difference between Single 0:04:26

Difference between Single and Dual Carriageway/Driving Lesson UK!

Linear Algebra - 0:18:53

Linear Algebra - Dual Space, Dual Basis, Double Dual

Linear Algebra: dual 0:52:11

Linear Algebra: dual space, double dual, dual basis and annihilator of subspace, 9-11-24

Roundabout Lane Discipline 0:10:56

Roundabout Lane Discipline | Learn to drive: Intermediate skills

How to enter 0:04:51

How to enter a Dual Carriageway via the acceleration lane(slip road)

Brian Cox explains 0:01:22

Brian Cox explains quantum mechanics in 60 seconds - BBC News

How the heart 0:04:28

How the heart actually pumps blood - Edmond Hui

Dual of Boolean 0:02:58

Dual of Boolean Expression

ROUNDABOUTS: Spiral & 0:06:59

ROUNDABOUTS: Spiral & Multi-lane Roundabouts Made Easy Part 3 - How to Choose the Correct Lane

How to do 0:01:30

How to do parallel parking - Driving lessons with AA Driving School

Unlocking the Secrets 0:12:08

Unlocking the Secrets of Arabic Grammar: Understanding Singular, Dual, and Plural'

Clutch, How does 0:06:47

Clutch, How does it work?

6:1 Dual Permeability 0:41:17

6:1 Dual Permeability - Dual Porosity Systems

Комментарии
Автор

0:26 Hahn-Banach recap
3:17 Hahn-Banach application: unit-norm in dual space
8:17 Double dual
24:35 Reflexive spaces
28:47 Lebesgue integration / outer measure
40:35 Actual start of Lebesgue integration
52:56 Outer measure
1:06:50 Outer measure additivity theorem

travischapman
Автор

The introduction to the Lebesgue integral and measure theory was really smooth. Good stuff!

TheTacticalDood
Автор

The norm is a real valued function, the range of the bounded linear functionals is the field of complex numbers. In your demonstration of boundedness, you are comparing the size of a real number to a complex number, which doesn't make sense. You need the magnitude of the complex number in front of the norm.

briang.valentine
Автор

In 50:41 the [INAUDIBLE] part is "due to 'Carathéodory'".

emir
Автор

7:35 I don't understand why does the inequality hold? Why the t has unit length?

meaningseeker
Автор

22:30 why |f(v)|<= ||T_v|| * ||f||?

HaozheJiang