Introduction To Functional Analysis: Metric Space

Learn the key concept of metric space in functional analysis with a lot of examples and solved exercises

What you’ll learn
What is metric space?
Cauchy’s Schwarz and Malinowski”s Inequality
Open balls, closed balls and open sets
The concepts of neighborhood and limit points
Interior and exterior points
Convergence of a sequence in a metric space
Many examples and exercises along with their solutions

Requirements
Pre Calculus, Calculus and Basics Algebra

Description
How to Become Pro in Functional Analysis: Metric Space?After the calculus, Functional analysis: metric space is one of the advanced courses in mathematics. Moreover, Functional analysis: metric space is actually comprised of three areas, like metric space, topology, and norm space. In this course of Functional analysis: metric space, we will understand metric space with definitions and examples. We will solve the various problems that define the metric space. This course approximately covers all the contents that are used in metric space.Most of the lectures are recorded on classroom whiteboards and a few lectures on a tablet. It is best practiced in this course to explain every concept according to the level of the subjects and students. Students will feel no difference between classroom learning and online learning. It is fully tried in this course to develop the connections between the students and the instructor.The length of this course is approximately 10 hours and all the videos have been recorded by the instructor in HD mode. The videos are edited after the recording to make them more attractive and visible for the students so that they can easily follow the method of instruction. However, the instructor is fully committed to answering the questions put by the students.The instructor’s accent is Asian and understandable for any English speaker. However, non-native English speakers can opt for captions.CONTENTS OF Functional AnalysisWhat is metric space?Cauchy’s Schwarz and Malinowski”s InequalityOpen balls closed balls and open setsThe concepts of neighborhood and limit pointsInterior and exterior pointsConvergence of a sequence in a metric spaceMany examples and exercises along with their solutions I like those students who ask the questions in the questions answer sections. It has come to my experience that the students who ask the questions. they learn more and they have a better understanding of the subject. So, you should not feel shy and you shouldn’t have any kind of hesitation to talk with your instructor.I will also be going to start the notes to my students on abstract algebra and by following the notes they can download the notes and they can even discuss the notes with me. The notes will be my own handwritten notes. And the notes will contain more materials than the videos.Functional Analysis is usually offered in different universities in different universities around the world in the mathematics department and the students of pure mathematics, must take this course and of courses, you should have maths skills to take this functional analysis.While taking this course I have some suggestions to my students that they must follow the following instruction while taking this course1. You must have one notebook and a pencil to note the important concepts. Just listening is distorted learning and this kind of learning in mathematics especially is not allowed. Every new step in mathematics has a link with the previous concepts. So, when you will note the important things in functional analysis then you will learn more and fast.2. You shouldn’t watch the course for more than one hour daily. So, in this way, you will complete the course in two weeks. By doing this you bind the functional analysis more accurately in your mind.3. You should watch this course more than 2 times and then many times as you have lifetime access to this course. You can repeat it each or every year. In this way, you will be a super genius in fictional analysis.4. Must share the course concepts with your friends of mathematics and if possible then group discussion is much helpful in this way. Knowledge increases by sharing. Sharing and practicing mathematics is only the one key to have mastery over the subject.5. Don’t skip any video while watching this course. Again I will say that this is distorted learning. Students have much practice by skipping the videos and I think this is the oddest way by taking an online course.6. Must keep in contact with your instructor and ask everything about the course. What is the advantage of the course and what they will get after this course? So, discussion with your course instructor is much important.Functional Analysis is a much easy and interesting subject. The mathematical steps in functional analysis are enjoyable and you will not make any boring situation while taking this course.Moreover, feedback is much necessary when you will finish the course. The future students who will enroll in this course will enroll on the basis of your feedback, therefore must give the feedback by writing the review or giving the stars. I will just say you finally must stay healthy and increase your existing skills by joining this course functional analysis. See you in the course.

Overview
Section 1: Introduction

Lecture 1 Introduction

Section 2: Metric Space

Lecture 2 Definition of Metric Space

Lecture 3 Usual Metric on R

Lecture 4 d is Metric on X

Lecture 5 d is Metric on X 2nd Example

Lecture 6 d is Metric on Set of Complex Number

Lecture 7 d is Metric on Set of Complex Number part 2

Lecture 8 d is metric on R^n

Lecture 9 d is Metric on R^n Part 2

Lecture 10 d is Metric on R^n Part 3

Lecture 11 Discrete Metric Example 1

Lecture 12 Discrete Metric Example 2

Lecture 13 d is not Metric on R

Lecture 14 d is Metric on R^2

Lecture 15 d is Metric on Set of All Continuous Function

Lecture 16 d is Metric on R^2

Lecture 17 d is Metric on Set of All Bounded Functions

Lecture 18 d is Metric on X

Lecture 19 d is Metric on X Part 2

Lecture 20 Another Metric on X define by a Function

Lecture 21 d is not Metric

Lecture 22 d is Metric on R^2

Lecture 23 d is not Metric

Lecture 24 Metric on Another Function

Lecture 25 An Inequality

Section 3: Cauchy’s Schwarz Inequality

Lecture 26 Cauchy’s Schwarz Inequality

Lecture 27 Cauchy’s Schwarz Inequality Part 2

Lecture 28 Minkowski’s Inequality

Section 4: Open Ball

Lecture 29 Open Ball

Lecture 30 Definition of Open Ball

Lecture 31 Examples of Open Ball

Lecture 32 Open Ball Theorems

Section 5: Closed Ball and Open Sets

Lecture 33 Definition and Examples of Closed Ball

Lecture 34 Open Sets and Theorem in Open Sets

Lecture 35 Open Sets Theorems

Lecture 36 Union of Open Sphere is an Open Set

Section 6: What is Complete Space?

Lecture 37 What is Complete Space?

Section 7: Convergence of a Sequence in a Metric Space

Lecture 38 Convergent Sequences

Lecture 39 Monotonic Sequence

Lecture 40 Examples and Exercises in Sequences

Lecture 41 Examples and Exercises in Sequences 2

Lecture 42 Examples and Exercises in Sequences 3

Lecture 43 Examples and Exercises in Sequences 4

Lecture 44 Examples and Exercises in Sequences 6

Lecture 45 Examples and Exercises in Sequences 7

Lecture 46 Examples and Exercises in Sequences 8

Lecture 47 Examples and Exercises in Sequences 9

Students which are interested in learning advance mathematics