Mathematics - Real Analysis (English)

   Thursday, March 27, 2025   

Starting a new Lecture Notes Series on Mathematics - Real Analysis

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Mathematics - Real Analysis By Lecture Notes together!

Lecture 1: Mod-01 Lec-1 Introduction

Lecture 2: Mod-01 Lec-02 Functions and Relations

Lecture 3: Mod-01 Lec-3 Finite and Infinite Sets

Lecture 4: Mod-01 Lec-4 Countable Sets

Lecture 5: Mod-01 Lec-5 Uncountable Sets, Cardinal Numbers

Lecture 6: Mod-02 Lec-06 Real Number System

Lecture 7: Mod-02 Lec-7 LUB Axiom

Lecture 8: Mod-02 Lec-08 Sequences of Real Numbers

Lecture 9: Mod-02 Lec-09 Sequences of Real Numbers - continued

Lecture 10: Mod-02 Lec-10 Sequences of Real Numbers - continued...

Lecture 11: Mod-02 Lec-11 Infinite Series of Real Numbers

Lecture 12: Mod-02 Lec-12 Series of nonnegative Real Numbers

Lecture 13: Mod-02 Lec-13 Conditional Convergence

Lecture 14: Mod-03 Lec-14 Metric Spaces: Definition and Examples

Lecture 15: Mod-03 Lec-15 Metric Spaces: Examples and Elementary Concepts

Lecture 16: Mod-03 Lec-16 Balls and Spheres

Lecture 17: Mod-03 Lec-17 Open Sets

Lecture 18: Mod-03 Lec-18 Closure Points, Limit Points and isolated Points

Lecture 19: Mod-03 Lec-19 Closed sets

Lecture 20: Mod-04 Lec-20 Sequences in Metric Spaces

Lecture 21: Mod-04 Lec-21 Completeness

Lecture 22: Mod-04 Lec-22 Baire Category Theorem

Lecture 23: Mod-05 Lec-23 Limit and Continuity of a Function defined on a Metric space

Lecture 24: Mod-05 Lec-24 Continuous Functions on a Metric Space

Lecture 25: Mod-05 Lec-25 Uniform Continuity

Lecture 26: Mod-06 Lec-26 Connectedness

Lecture 27: Mod-06 Lec-27 Connected Sets

Lecture 28: Mod-06 Lec-28 Compactness

Lecture 29: Mod-06 Lec-29 Compactness - Continued

Lecture 30: Mod-06 Lec-30 Characterizations of Compact Sets

Lecture 31: Mod-06 Lec-31 Continuous Functions on Compact Sets

Lecture 32: Mod-06 Lec-32 Types of Discontinuity

Lecture 33: Mod-07 Lec-33 Differentiation

Lecture 34: Mod-07 Lec-34 Mean Value Theorems

Lecture 35: Mod-07 Lec-35 Mean Value Theorems - Continued

Lecture 36: Mod-07 Lec-36 Taylor's Theorem

Lecture 37: Mod-07 Lec-37 Differentiation of Vector Valued Functions

Lecture 38: Mod-08 Lec-38 Integration

Lecture 39: Mod-08 Lec-39 Integrability

Lecture 40: Mod-08 Lec-40 Integrable Functions

Lecture 41: Mod-08 Lec-41 Integrable Functions - Continued

Lecture 42: Mod-08 Lec-42 Integration as a Limit of Sum

Lecture 43: Mod-08 Lec-43 Integration and Differentiation

Lecture 44: Mod-08 Lec-44 Integration of Vector Valued Functions

Lecture 45: Mod-08 Lec-45 More Theorems on Integrals

Lecture 46: Mod-09 Lec-46 Sequences and Series of Functions

Lecture 47: Mod-09 Lec-47 Uniform Convergence

Lecture 48: Mod-09 Lec-48 Uniform Convergence and Integration

Lecture 49: Mod-09 Lec-49 Uniform Convergence and Differentiation

Lecture 50: Mod-09 Lec-50 Construction of Everywhere Continuous Nowhere Differentiable Function

Lecture 51: Mod-09 Lec-51 Approximation of a Continuous Function by Polynomials: Weierstrass Theorem

Lecture 52: Mod-09 Lec52 Equicontinuous family of Functions: Arzela - Ascoli Theorem