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Functional Analysis (Part-1)
23 students

Functional Analysis (Part-1)

Complete Introduction to Metric Spaces
Created byWasif Ahmed
Last updated 3/2025
English
English [Auto],

What you'll learn

  • Complete understanding of Metric spaces
  • Both Basic and Advanced Topics of Metric Spaces
  • Cruise into Functional Analysis

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Course content

2 sections • 13 lectures • 5h 20m total length

  • Definition and brief concepts26:00

    In this video the student will understand the definition of metric space and the properties of a metric.  Moreover, the triangle inequality is generalized. In addition, it is also illustrated how to induce a metric on a restriction of a given set.

  • Example of Metric Space27:51

    Here you will have an idea of how to prove a given space with a given metric a metric space.

  • More Examples of Metric space48:17

    More examples are included : sequence space ,function space and discrete metric space

  • Student Exercises-110:06

    Explore three- and n-dimensional Euclidean spaces with the Euclidean metric, and extend to unitary (complex) spaces, proving these metrics yield metric spaces and understanding the complex plane.

  • Student Exercises-2
  • Solutions to Student Exercise-2 (part1)23:58

    This lecture tests when the real line's distance is a metric, shows squared distance fails the triangle inequality, and proves sqrt(|x - y|) is a valid metric.

  • Solution to Student Exercise-2 (part 2)36:34

    Explore metrics on sequence spaces and function spaces: derive the discrete metric on subspaces of binary sequences, verify the integral metric on continuous functions, and analyze the Hamming distance on {0,1}^3.

  • Solutions to Student Exercise-2 (part3)20:42

    Explore the triangle inequality and generalized triangle inequality within metric spaces, proving non-negativity and equivalence of metric axioms using foundational inequalities.

  • Text Book0:03

Requirements

  • A basic knowledge of math is enough

Description

This course Functional Analysis (part1) mainly focuses on Metric spaces. It provides a good material to understand the ideal of metric spaces to start Functional Analysis-2. The Book followed for this course is the famous text on Functional Analysis and its Applications written by Erwin Kreyszig.

Main contents of the course are as follows:

Definition of Metric Spaces with Properties

Some Basic Ideas including, induced metric, generalized triangle inequality

Basic Examples of Metric Space

      The usual metric space

      The Euclidean Plane ℝ²

      The Euclidean Space ℝ³

      The Function Space C[a,b]

      The Space ℓ∞ (L-infinity)

      The Space Lᵖ

      The Space B(A) of Bounded Functions

      The Sequence Space 's'

Some useful Inequalities are derived

       Holder Inequality

       Minkowski Inequality

       Cauchy - Schwarz Inequality 

       An Auxiliary Inequality

Student Exercises with solutions are explained.

Following topics are being worked (They are uploaded in Hindi/Urdu with english subtitles, soon they will be available in English)

Separable spaces

Complete metric spaces


All of the sections, Examples and Exercises from the first chapter of this book are explained by me in simple way. i hope the students will learn much from my video lectures and will give me positive feed back. After completing this course the students are recommended to take Functional Analysis-2 in which second chapter of the textbook is explained in videos.

Who this course is for:

  • This course is taught to the undergraduate students of Mathematics, Physics and Engineering.

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