Real Analysis

Real Analysis and Theory of Convergence

This book includes the following topics

Real Number System

• Introduction
• Field axiom
• Uniqueness property
• Cancellation law of addition and multiplication
• Order axiom and ordered field
• Positive class
• Boundedness

Upper bound, Supremum, Lower bound, Infimum, Bounded set

• Greatest and least element
• Completeness axiom
• Complete ordered field
• Archimedean property of real numbers

Archimedean ordered field

• Betweenness theorem
• Dedekind’s completeness axiom
• Irrational numbers
• Rational density theorem or denseness property
• Absolute value of a real number or Modulus

Point Set Topology

• Introduction
• Neighbourhood (nbd) of a real number
• Properties of neighbourhood
• Interior and exterior point of a set
• Interior of a set
• Open set
• Limit point of a set
• Derived set and closed set
• Closure
• Open and closed interval
• Nested interval property
• Bolzano-Weierstrass theorem
• Complement of set
• Open cover, subcover and compact set
• Heine Borel theorem
• Connected and disconnected set

Countable Sets

• Introduction
• Equivalent sets
• Finite and infinite set
• Countable set
• Uncountable set
• Cantor ternary set
• Binary representation
• Ternary representation
• Construction of Cantor ternary set
• Properties of Cantor ternary set

Real Sequences

• Introduction
• Sequence
• Range of a sequence
• Bounded and unbounded sequence
• Supremum and infimum of sequence
• Monotonic sequence
• Limit point of a sequence
• Bolzano-Weierstrass theorem
• Limit of a sequence
• Convergent sequence
• Divergent sequence
• Oscillatory sequence
• Theorems on convergence sequences
• Theorems on convergence of monotonic sequences
• Algebra of sequences
• Sandwich theorem
• Limit superior and limit inferior
• Sub-sequence
• Some theorems of sub-sequence
• Cauchy’s sequence or fundamental sequence
• Some important theorems of Cauchy’s sequence
• Cauchy’s general principle of convergence for sequence
• Cauchy’s first theorem on limits
• Cauchy’s second theorem on limits
• Cesaro’s theorem

Infinite Series

• Introduction
• Sequence of partial sums of series
• Nature of an infinite series
• Some important theorems

Cauchy’s general principle of convergence
Test of the convergence of geometric series

• Comparative tests of the first type
• Comparative tests of the second type

Ratio-comparison test; D’ Alembert’s ratio test; Raabe’s test; de Morgan’s and Bertrand’s test; Logarithmic ratio test; Second logarithmic ratio test; Gauss’s test

• Some other useful tests

Cauchy’s n^th root test; Cauchy’s condensation test

• Alternating series
• Absolute convergence
• Conditionally convergence

Uniform Convergence

• Introduction
• Pointwise convergence of a sequence of functions
• Uniform convergence
• Series of functions
• Cauchy’s criterion for uniform convergence
• Test for uniform convergence of a sequence and series of functions
• Uniform convergence and continuity
• Term by term integration
• A sufficient condition for term by term integration of an infinite series
• A sufficient condition for term by term differentiation of the series

Improper Integrals

• Finite and infinite intervals
• Bounded function
• Improper integral
• Types of improper integral
• Convergence of improper integral of first kind
• Convergence tests for the improper integral of first kind
• Convergence of improper integral of second kind
• Convergence tests for the improper integral of second kind
• Convergence of improper integrals of third kind

Riemann Integration

• Introduction
• Partition of a closed interval
• Norm of partition
• Refinement of a partition
• Supremum and infimum
• Upper and lower Darboux sum
• Theorems on Darboux sum
• Upper and lower Riemann integral
• Integral function
• Riemann integral
• Theorems of Riemann integral

Necessary and sufficient condition for a function to be R-integrable

• Particular classes of Riemann integrable functions
• Riemann integral as the limit of a sum
• Properties of Riemann integral function
• Integral function
• Properties of integral function
• Primitive
• Mean value theorems of integral calculus
• Fundamental theorem of integral
• Techniques of integration

Fourier Series

• Introduction
• Perodic functions
• Properties of definite integral
• Some important definite integrals
• Fourier series
• Dirichlet’s conditions for the expansion of a Fourier series
• Even and odd functions
• Fourier series for even and odd functions
• Fourier’s half range series
• Other forms of Fourier series