Real Analysis
Good Books for reference:
Introduction to analysis (Bartley and Sherbert)
Principles of mathematical analysis (W. Rudin)
Mathematical analysis (Apostol)
Syllabus for real analysis
Elementary set theory
Finite, countable and uncountable sets
Real number system as a complete ordered field
Archimedean property
Supremum, Infimum
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Sequences and series
Convergence
Lecture 3 Behaviour of sequences.
Lecture 4: Epsilon N definition of Limits of sequences
Lecture 5: Recursive definition of sequences.
Lecture 6: What are subsequences?
Lecture 7: Nondecreasing sequences.
Lecture 8: Bounded sequences
Lecture 9: Bounded and Nondecreasing sequences are convergent
Lecture 10: Properties of Limits of sequences
Lecture 11: Frequently arising Limits
Lecture 12: Introduction to series.
Lecture 13: Checking the convergence of a series by the sequence of partial sums
Lecture 14: Geometric series
Lecture 15: Some Problems based on geometric series
Lecture 16: Telescopic series
Lecture 17: nth term test for checking the divergence of a series.
Lecture 18: Some noteworthy points regarding convergence of series.
Lecture 19: The integral test for checking the convergence of series of positive terms
Lecture 20: Problems based on integral test.
Lecture 21: p test for checking convergence/divergence of a series
Lecture 22: Comparison tests ( Direct comparison test and limit comparison test ).
Lecture 23: Problems on Limit comparison test (LCT) and Direct comparison test (DCT)
Lecture 24: Ratio test
Lecture 25: Nth root test.
Lecture 26: Alternating series
limsup, liminf
Bolzano Weierstrass theorem
Heine Borel theorem
Continuity
Uniform continuity
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Differentiability
Mean value theorem
Lecture 1: What is a function?
Lecture 2: min-max theorem for continuous functions.
Lecture 3: Local and Global extreme values.
Lecture 4: First derivative theorem for local extreme values.
Lecture 5: Problems on finding extreme values of functions.
Lecture 6: All critical and boundary points may not be the points of local extreme values.
Lecture 7: How to find out if a critical/boundary point is a point of local extreme value?
Lecture 8: Problems on finding extreme values revisited
Lecture 9: Rolle’s theorem
Lecture 10: The mean value theorem
Lecture 11: Concave up and concave down graphs
Lecture 12: Point of inflection
Lecture 13: Cartesian graphing using first and second derivatives I
Lecture 14: Cartesian graphing using first and second derivatives-II
Lecture 15: Cartesian graphing using first and second derivatives-III
Sequences and series of functions
Uniform convergence
Riemann sums and Riemann integral
Improper Integrals
Monotonic functions
Types of discontinuity
Functions of bounded variation
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Lebesgue measure
Measure theory 1
Measure theory 2
Measure theory 3
Measure theory 4 (The Lebesgue outer measure is translation invariant)
Measure theory 5 (Outer Lebesgue measure is countably/finitely subadditive.)
Measure theory 6 (Outer measure of an interval is equal to its length.)
Measure theory 7 (What is a Lebesgue measurable subset?)
Measure theory 8 (Why the Lebesgue measurable set is defined in the way it is defined?)
Measure theory 9 (Properties of Lebesgue measurable sets.)
Measure theory 10 (Countable sets are Lebesgue measurable sets.)
Measure theory 11 (Set of Lebesgue measurable subsets is a sigma algebra.)
Measure theory 12 (Finite union of Lebesgue measurable sets is a Lebesgue measurable set.)
Measure theory 13 (Finite union of Lebesgue measurable sets is Lebesgue measurable set.)
Measure theory 14 (Countable union of Lebesgue measurable sets is a Lebesgue measurable set.)
Measure theory 15 (Countable additivity of Lebesgue outer measure for LMS.)
Measure theory 16 (Every interval is measurable.)
Measure theory 17 (open, closed,G_delta,F_sigma,Borel sets are measurable.)
Measure theory 18 (Translation of a Lebesgue measurable set is a Lebesgue measurable set.)
Measure theory 19 (Summary on the properties of sigma algebra of all Lebesgue measurable sets.)
Measure theory 20 (Lebesgue measure to Lebesgue integral.)
Measure theory 21 (Riemann integral)
Measure theory 22 ( Example of the function which is not Riemann integrable)
Measure theory 23 ( Drawback of Riemann integration. )
Measure theory 24 ( Exchanging limit and integral sign for Riemann integral )
Measure theory 25 (Outer approximations of Lebesgue measurable sets with open and Gdelta setsI)
Measure theory 26 (Outer approximations of Lebesgue measurable sets with open and Gdelta sets-II)
Measure theory 27 (Inner approximations of Lebesgue measurable sets with closed and F-sigma sets)
Measure theory 28 : Approximations of Lebesgue measurable sets of finite measure
Measure theory 29 (Continuity of Lebesgue measure)
Measure theory 30 (Borel Cantelli Lemma)
Measure theory 31 (Non Lebesgue measurable set I)
Measure theory 32 Non Lebesgue measurable set II
Measure theory 33 ( Non Lebesgue measurable set III)
Measure theory 34 (Non Lebesgue measurable set IV Vitali theorem)
Measure theory 35 (Can we explicitly construct non Lebesgue measurable set?)
Measure theory 36 (Lebesgue measurable function)
Measure theory 37 (Why Lebesgue measurable functions are defined so?)
Measure theory 38 (Properties of Lebesgue measurable functions)
Measure theory 39 Monotonic functions defined on an interval are LMF
Measure theory 40 (Sum and product of Lebesgue measurable functions)
Measure theory 41 (Sum and product of Lebesgue measurable functions ii)
Measure theory 42 (Sum and product of Lebesgue measurable functions -iii)
Measure theory 43 (Division of two Lebesgue measurable functions is a Lebesgue measurable function)
Measure theory 44 (What about composition of two Lebesgue measurable functions?)
Measure theory 45 (Minima and maxima of a finite family of Lebesgue measurable functions is a LMF)
Measure theory 46 (Function as a difference of two Lebesgue measurable functions)
Measure theory 47 (Pointwise and uniform convergence of a sequence of functions)
Measure theory 48 (Pointwise and uniform convergence of a sequence of functions II)
Measure theory 49 (Uniform convergence is better than pointwise convergence)
Measure theory 50 (Lebesgue measurability is preserved in pointwise convergence)
Measure theory 51 (Characterstic function, existence of non Lebesgue measurable functions)
Measure theory 52 (Simple functions)
Measure theory 53 (Simple approximation lemma)
Measure theory 54 (Simple approximation theorem)
Measure theory 55 (Littlewood's first principle)
Lebesgue integral
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Functions of several variables
Lecture 1: Functions of several variables
Lecture 2: Domain and Range of Functions of more than one variables
Lecture 3: Open and closed sets
Lecture 4: Graph; Level curves; Contour lines of functions of two variables
Lecture 5: Limits of functions of two variables-I
Lecture 6: Problems based on epsilon delta definition of Limits
Lecture 7: Properties of Limits of functions of two variables
Lecture 8: Two path method to declare that limit does not exist
Lecture 9: Continuity of functions of two variables
Directional derivative
Partial derivative
Derivative as a linear transformation
Inverse and implicit function theorems
Metric spaces
Compactness
Connectedness
Normed linear Spaces
Spaces of continuous functions as examples
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