University Mathematics · Original Course Resource

Real Analysis II — Complete Course Notes, Proofs & Solved Problems

Real Analysis-II · Real Analysis 2 · Analysis II · Advanced Real Analysis

One progressively developed course resource covering rigorous integration, Riemann-Stieltjes theory, bounded variation, improper integrals, uniform convergence, power and Fourier series, Euclidean spaces, multivariable analysis and multiple Riemann integration.

Prepared by Mehreen Kanwal — MPhil Mathematics · Rana Ali Hasan — MPhil Mathematics.

Associated academic brand: The Science Academy, Chak No. 84 NB, Sargodha.

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$Real Analysis II complete course notes and solved mathematics book by Mehreen Kanwal and Rana Ali Hasan — The Math Hub$

About this book

A continuous route from integration to multivariable analysis

Real Analysis II on The Math Hub begins with rigorous Riemann integration and advances through Riemann-Stieltjes theory, bounded variation, improper integrals, uniform convergence, power series, approximation, Fourier series, Euclidean-space analysis, multivariable differentiation and multiple Riemann integration. It is intentionally broader than a single university outline so that students can use one coherent book while studying overlapping Real Analysis-II syllabi across Pakistani universities.

Definitions, hypotheses and intermediate mathematical steps are kept visible. The course includes zero-to-advanced worked examples, proof pathways and exercises; it does not reproduce long copyrighted reference-book exercises.

20numbered chapters

543A4 pages in the verified master PDF

Riemann → Lebesgueadvanced bridge, not a full Measure Theory course

Englishcourse notes and book resource

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Why study this course?

What Real Analysis II makes precise

Rigor behind calculus

Real Analysis supplies the rigorous foundations behind limits, continuity, differentiation and integration.

Safe interchange of operations

Uniform convergence explains when limits, sums, derivatives and integrals may be interchanged safely.

Bridges to advanced mathematics

Approximation, Fourier series, R^n, topology, functional analysis, differential geometry, optimization and measure theory appear as connected next steps.

Purpose and benefits

From definitions to usable proof methods

The resource builds the habit of stating hypotheses, selecting a theorem, showing intermediate estimates, checking counterexamples and interpreting the result. This makes it useful for examination preparation as well as later analysis courses.

Generalize ordinary integration through Riemann-Stieltjes and bounded variation.
Control parameter-dependent limits, series, integrals and derivatives.
Use power series, Taylor theory, Fourier methods and polynomial approximation.
Move rigorously from the real line to R^n and multiple integration.

Who this resource is for

Relevant learners and programmes

BS Mathematics students studying Real Analysis II / Analysis II.
University students whose syllabus includes Riemann-Stieltjes integration, improper integrals or uniform convergence.
Legacy BSc / MSc Mathematics students revising classical Real Analysis.
ADP learners where advanced calculus / analysis overlaps these topics.
MS / MPhil entrants needing a rigorous undergraduate refresher.
Independent learners and students preparing for proof-based university mathematics.

Learning outcomes

By the end of the route, a learner can:

Formulate and use rigorous definitions of Riemann and Riemann-Stieltjes integrability.
Work with functions of bounded variation and connect variation with Stieltjes integration.
Test convergence of improper and parameter-dependent improper integrals.
Use Beta and Gamma functions and their main identities.
Distinguish pointwise and uniform convergence and apply uniform-convergence criteria.
Determine when limits, sums, integrals and derivatives may be interchanged.
Analyze power series and Taylor/Maclaurin expansions.
Understand equicontinuity, compact families and approximation theorems.
Compute and analyze Fourier coefficients and Fourier-series convergence.
Work rigorously in Euclidean spaces R^n and with functions of several variables.
Use gradients, Jacobians, Hessians and differentiability in R^n.
Solve constrained/unconstrained extrema problems and use multiple Riemann integrals.
Understand why Riemann theory motivates the transition to measure and Lebesgue integration.

Course contents · exactly 20 entries

The complete chapter sequence

Each entry opens a separate crawlable chapter block below. The Advanced Bridge is kept separately from the 20 numbered chapters.

Chapter 01: Riemann Integration: Rigorous FoundationsPartitions, Darboux sums, integrability criteria, Riemann sums and the Fundamental Theorem of Calculus
Chapter 02: The Riemann-Stieltjes IntegralIntegration with respect to an increasing function, existence theorems, jumps, weights and transformation rules
Chapter 03: Functions of Bounded VariationTotal variation, algebraic structure, Jordan decomposition, Stieltjes integrators and rectifiable curves
Chapter 04: Improper IntegralsInfinite intervals, unbounded integrands, convergence criteria, absolute and conditional convergence, Dirichlet-Abel methods and tail estimates
Chapter 05: Uniform Convergence of Improper IntegralsParameter-dependent integrals, uniform Cauchy criteria, Weierstrass domination, continuity, limit interchange, differentiation under the integral sign and uniform Dirichlet-Abel tests
Chapter 06: Beta and Gamma FunctionsEuler's gamma and beta integrals, convergence, recurrence, special values, beta-gamma relation, Gaussian integral, transformations and applications
Chapter 07: Sequences of FunctionsFunction sequences, pointwise and uniform convergence, uniform Cauchy criterion, sup-norm control, algebraic stability, counterexamples and local uniform convergence
Chapter 08: Uniform Convergence and ContinuityUniform limits of continuous functions, epsilon proofs, compactness, uniform continuity, Dini's theorem, sup-norm completeness and diagnostic counterexamples
Chapter 09: Series of FunctionsPartial sums, pointwise and uniform convergence, uniform Cauchy criterion, Weierstrass M-test, Dirichlet and Abel tests, alternating estimates, rearrangements and products
Chapter 10: Interchanging Limits, Sums, Integrals and DerivativesUniform convergence as the main mechanism for passing limits through continuity, Riemann integration and differentiation; termwise operations, parameter limits and sharp counterexamples
Chapter 11: Power SeriesConvergence geometry, radius of convergence, Cauchy--Hadamard theory, uniform convergence on compact subintervals, endpoint analysis and termwise operations
Chapter 12: Taylor and Maclaurin SeriesTaylor polynomials, remainder formulas, rigorous error estimates, analyticity, standard expansions and approximation by local power series
Chapter 13: Equicontinuity and Compact Families of FunctionsUniform control of entire families, boundedness, equicontinuity, Arzela--Ascoli compactness and subsequence methods in spaces of continuous functions
Chapter 14: Polynomial ApproximationWeierstrass approximation, Bernstein polynomials, density in continuous-function spaces, Stone--Weierstrass theory and uniform approximation methods
Chapter 15: Fourier SeriesOrthogonal trigonometric systems, Fourier coefficients, Bessel inequality, convergence theorems, half-range expansions, best mean-square approximation and Parseval theory
Chapter 16: Euclidean Space and Metric FoundationsEuclidean norm, distance, open and closed sets, limit points, compactness, connectedness, sequences and the Heine--Borel theorem
Chapter 17: Limits and Continuity in Several VariablesMultivariable limits, epsilon--delta and sequential criteria, path tests, continuity, compact-domain consequences and vector-valued maps
Chapter 18: Differentiability in R^nPartial and directional derivatives, total derivative, gradient, Jacobian matrix, differentiability criteria, chain rule and mean-value estimates
Chapter 19: Higher Derivatives, Taylor Theory and ExtremaSecond derivatives, Hessian matrix, mixed partials, multivariable Taylor theorem, unconstrained and constrained extrema, Lagrange multipliers, implicit and inverse-function bridges
Chapter 20: Multiple Riemann IntegrationRiemann integration on rectangles in R^n, Jordan content, iterated integrals, Fubini-type theorems, general regions, Jacobians and change of variables

Advanced Bridge: From Riemann to LebesgueConceptual transition only · full Measure Theory remains separate

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Detailed academic coverage

Chapter-by-chapter definitions, theorems, methods and applications

Every chapter keeps its genuine topic names visible and links to the complete owner PDF rather than inventing chapter downloads.

Chapter 1

Chapter 1: Riemann Integration: Rigorous Foundations

Chapter scope: Partitions, Darboux sums, integrability criteria, Riemann sums and the Fundamental Theorem of Calculus

What is taught

Why integration needs partitions
Refinement
Local upper and lower bounds
Geometric interpretation
Definition of Darboux sums
Refinement monotonicity
Lower and upper Darboux integrals
Formal definition
Oscillation criterion
Cauchy-type criterion using tagged sums
A single discontinuity can be harmless
Continuous functions are integrable
Monotone functions are integrable
Algebraic closure
Order and integral inequalities
Tagged partitions
Endpoint sums
The integral as an accumulation function
Change of variables
Integration by parts

Definitions and foundational objects

Partition
Norm or mesh of a partition
Refinement
Lower and upper Darboux sums
Lower and upper Darboux integrals
Riemann integrability in the Darboux sense
Tagged partition
Riemann integral via tagged sums

Important theorems, rules and methods

Refinement improves Darboux bounds
Every lower sum is below every upper sum
Fundamental inequality
Darboux criterion for integrability
Continuous functions on closed intervals are Riemann integrable
Monotone integrability
Algebra of Riemann integrable functions
Equivalence of Darboux and Riemann definitions
Continuity of the indefinite integral
Fundamental Theorem of Calculus - Part I
Fundamental Theorem of Calculus - Part II (Newton-Leibniz)
Substitution
Integration by parts

Why this chapter is taught: Build rigor. Understand integrability. Connect limits with area. Explain the main definitions and notation used in the chapter.

Worked examples, proofs and exercises: The complete book uses zero-to-advanced worked examples and proof-based exercises for this chapter. Study definitions → lemmas or intermediate estimates → theorem statement → complete proof pathway → corollaries/counterexamples → worked examples and exercises without replacing the supplied book with generic one-line examples.

Applications and connections: calculus foundations · numerical integration · rigorous integral theory.

Learning outcomes / benefits

Explain the main definitions and notation used in the chapter.
Follow the chapter's proofs without skipping hypotheses or intermediate mathematical steps.
Apply the core results to worked examples.
Recognize counterexamples and understand why theorem assumptions matter.
Use the chapter as preparation for later Real Analysis, Measure Theory, Functional Analysis or related advanced courses.

Chapter 2

Chapter 2: The Riemann-Stieltjes Integral

Chapter scope: Integration with respect to an increasing function, existence theorems, jumps, weights and transformation rules

What is taught

The central idea
Ordinary Riemann integration as a special case
Simple weighted examples
Telescoping of integrator increments
Stieltjes-Darboux sums
Lower and upper Stieltjes integrals
Fundamental inequality
Refinement theorem
Oscillation form of the criterion
A warning about common discontinuities
Continuous integrands
Monotone integrands with continuous integrator
Linearity in the integrand
Linearity in the integrator
Additivity over intervals
Order and norm estimates
Unit jump integrator
Several jumps
Continuous plus discrete weighting
Reduction to an ordinary weighted integral
Integration by parts
Change of variable

Definitions and foundational objects

Stieltjes increment
Riemann-Stieltjes integrator
Lower and upper Riemann-Stieltjes sums
Lower and upper Riemann-Stieltjes integrals

Important theorems, rules and methods

Three basic cases
Refinement of Stieltjes sums
Cauchy-Darboux criterion for Stieltjes integrability
Equivalent criterion
Continuous integrand, increasing integrator
Monotone integrand, continuous increasing integrator
Linearity in f
Additivity
Order property
Integral against a single jump
Differentiable integrator
Riemann-Stieltjes integration by parts
Change of variable

Why this chapter is taught: Generalize integration. Track weighted change. Understand the integrator. Explain the main definitions and notation used in the chapter.

Applications and connections: probability-style weighted integration · distribution functions · bounded variation · measure-theory motivation.

Learning outcomes / benefits

Explain the main definitions and notation used in the chapter.
Follow the chapter's proofs without skipping hypotheses or intermediate mathematical steps.
Apply the core results to worked examples.
Recognize counterexamples and understand why theorem assumptions matter.
Use the chapter as preparation for later Real Analysis, Measure Theory, Functional Analysis or related advanced courses.

Chapter 3

Chapter 3: Functions of Bounded Variation

Chapter scope: Total variation, algebraic structure, Jordan decomposition, Stieltjes integrators and rectifiable curves

What is taught

Why boundedness is not enough
Geometric meaning
Monotone functions
A bounded derivative is a sufficient condition
Continuously differentiable functions
Step functions
A continuous bounded function that is not BV
BV functions are bounded
Linear combinations
Products
Reciprocals and quotients
Additivity of variation
The accumulated variation function
Jordan decomposition
Why the decomposition matters
Positive and negative variation
Continuity of the variation function
One-sided limits
Discontinuities of BV functions
Extending the integrator class
Rectifiable curves
Graphs of BV functions

Definitions and foundational objects

Variation
Total variation
Function of bounded variation
Variation function
Positive and negative variation
Rectifiable curve

Important theorems, rules and methods

Refinement increases variation sums
Monotone functions are of bounded variation
Bounded derivative implies bounded variation
Variation of a C^1 function
Bounded variation implies boundedness
Vector-space properties
Product of BV functions
Reciprocal
Additivity on adjacent intervals
T_f is increasing
Jordan decomposition for BV functions
Continuity is shared by f and its variation function
Regulated behaviour of BV functions
Countability of discontinuities
Continuous integrands and BV integrators
Variation estimate
Length of a C^1 curve

Why this chapter is taught: Measure change. Decompose oscillation. Connect variation with length. Explain the main definitions and notation used in the chapter.

Applications and connections: rectifiable curves · Stieltjes integrators · decomposition of functions · Fourier/analysis regularity.

Learning outcomes / benefits

Explain the main definitions and notation used in the chapter.
Follow the chapter's proofs without skipping hypotheses or intermediate mathematical steps.
Apply the core results to worked examples.
Recognize counterexamples and understand why theorem assumptions matter.
Use the chapter as preparation for later Real Analysis, Measure Theory, Functional Analysis or related advanced courses.

Chapter 4

Chapter 4: Improper Integrals

Chapter scope: Infinite intervals, unbounded integrands, convergence criteria, absolute and conditional convergence, Dirichlet-Abel methods and tail estimates

What is taught

Why a new definition is needed
The fundamental p-integral at infinity
Elementary algebra of convergent improper integrals
Singularities at an endpoint
The p-integral near zero
Mixed improper integrals
Cauchy criterion at infinity
Direct comparison for nonnegative integrands
Limit comparison
Comparison near a finite singularity
Absolute convergence
Dirichlet's test for improper integrals
Abel's test
Tail integrals
Change of variables in improper integrals
From integrals to series
Parameter limits: a preview
Extending the Stieltjes integral to an infinite interval
Reduction when the integrator is differentiable
Step integrators and infinite sums
A family of model convergence problems

Definitions and foundational objects

Two standard forms
Improper integral of the first kind
Integral over the whole real line
Improper integral of the second kind at the left endpoint
Interior singularity
Absolute convergence of an improper integral
Conditional convergence
Improper Riemann-Stieltjes integral at infinity

Important theorems, rules and methods

p-test at infinity
Linearity
p-test near a finite singularity
Local nature of improper convergence
Cauchy criterion for an infinite integral
Direct comparison
Limit comparison
Absolute convergence implies convergence
Dirichlet test
Abel test for improper integrals
Basic tail bound under absolute comparison
Integral test for positive decreasing functions

Why this chapter is taught: Extend definite integration beyond bounded intervals and bounded integrands. Explain the main definitions and notation used in the chapter.

Applications and connections: special functions · transforms · asymptotics · differential equations.

Learning outcomes / benefits

Explain the main definitions and notation used in the chapter.
Follow the chapter's proofs without skipping hypotheses or intermediate mathematical steps.
Apply the core results to worked examples.
Recognize counterexamples and understand why theorem assumptions matter.
Use the chapter as preparation for later Real Analysis, Measure Theory, Functional Analysis or related advanced courses.

Chapter 5

Chapter 5: Uniform Convergence of Improper Integrals

Chapter scope: Parameter-dependent integrals, uniform Cauchy criteria, Weierstrass domination, continuity, limit interchange, differentiation under the integral sign and uniform Dirichlet-Abel tests

What is taught

A new kind of convergence
Tail formulation
Uniform convergence near a finite singular endpoint
The uniform Cauchy principle
Why tail control is stronger than pointwise control
A useful monotone-tail criterion
Uniform Cauchy criterion near a singular endpoint
A parameter-independent majorant
Local domination near a singular endpoint
Parameter-restricted majorants
Dominating derivatives as well as functions
Uniform convergence preserves parameter continuity
Interchanging a parameter limit and an improper integral
What can go wrong without uniformity
The problem
Proof idea
Feynman-type parameter method
Repeated differentiation
Why absolute domination is not the whole story
Uniform Abel principle
Families with singular endpoints
A practical decision procedure

Definitions and foundational objects

Uniform tail condition
Uniform convergence of an improper integral

Important theorems, rules and methods

Uniform Cauchy criterion at infinity
Tail domination principle
Weierstrass test for improper integrals
Continuity under a uniformly convergent improper integral
Passage of a parameter limit through the integral
Differentiation under the integral sign
Useful sufficient domination condition
Uniform Dirichlet test, one useful form
Synthesis Principle

Why this chapter is taught: Control infinite tails uniformly in a parameter before interchanging analytical operations. Explain the main definitions and notation used in the chapter.

Applications and connections: parameter-dependent integrals · continuity/differentiation under the integral sign · applied analysis.

Learning outcomes / benefits

Explain the main definitions and notation used in the chapter.
Follow the chapter's proofs without skipping hypotheses or intermediate mathematical steps.
Apply the core results to worked examples.
Recognize counterexamples and understand why theorem assumptions matter.
Use the chapter as preparation for later Real Analysis, Measure Theory, Functional Analysis or related advanced courses.

Chapter 6

Chapter 6: Beta and Gamma Functions

Chapter scope: Euler's gamma and beta integrals, convergence, recurrence, special values, beta-gamma relation, Gaussian integral, transformations and applications

What is taught

Motivation: extending the factorial
Convergence near zero
Convergence at infinity
Positivity and smooth dependence on the parameter
The fundamental recurrence
Factorials appear automatically
The half-integer anchor
Extension beyond positive arguments: a preview
Definition
Symmetry
A useful infinite-interval representation
Trigonometric representation
The central identity
Proof by a double integral
Recurrences for the Beta function
Power-exponential integrals
Gaussian-power integrals
Rational integrals on [0,infinity)
Trigonometric integrals
The Gaussian integral and Gamma(1/2)
Legendre duplication formula
Parameter derivatives and logarithmic integrals
Log-convexity: an advanced structural property
Phase II synthesis

Definitions and foundational objects

Euler integrals
Gamma function
Beta function

Important theorems, rules and methods

Domain of Euler's gamma integral
Gamma recurrence
Factorial extension
Half-integer values
Convergence domain of the beta integral
Symmetry of the Beta function
Beta integral on [0,infinity)
Beta-Gamma relation
Beta recurrences
General gamma substitution
Gaussian and half-gamma values
Legendre duplication formula

Why this chapter is taught: Turn improper integrals into a reusable theory of special functions. Explain the main definitions and notation used in the chapter.

Applications and connections: probability distributions · special functions · integral evaluation · mathematical physics.

Learning outcomes / benefits

Explain the main definitions and notation used in the chapter.
Follow the chapter's proofs without skipping hypotheses or intermediate mathematical steps.
Apply the core results to worked examples.
Recognize counterexamples and understand why theorem assumptions matter.
Use the chapter as preparation for later Real Analysis, Measure Theory, Functional Analysis or related advanced courses.

Chapter 7

Chapter 7: Sequences of Functions

Chapter scope: Function sequences, pointwise and uniform convergence, uniform Cauchy criterion, sup-norm control, algebraic stability, counterexamples and local uniform convergence

What is taught

What changes when each term is a function?
How to find a pointwise limit
Pointwise boundedness
One index must work for the whole domain
The uniform error function
Uniform convergence on a smaller interval
Cauchy without knowing the limit
Sup norm
Completeness viewpoint
The moving-spike mechanism
Exponential concentration
A diagnostic principle
Linear combinations
Products require boundedness control
Reciprocals and quotients
Uniformity depends on the domain
Uniform convergence on unbounded domains
Compact-uniform error estimates

Definitions and foundational objects

Pointwise versus uniform
Sequence of functions
Pointwise convergence
Pointwise bounded sequence
Uniform convergence
Uniformly Cauchy sequence of functions
Supremum norm
Locally uniform convergence

Important theorems, rules and methods

Uniform convergence implies pointwise convergence
Supremum criterion for uniform convergence
Uniform Cauchy criterion
Sequential test for failure of uniform convergence
Linear stability
Product stability
Reciprocal stability

Why this chapter is taught: Control an entire family of errors with one index. Explain the main definitions and notation used in the chapter.

Applications and connections: approximation · numerical methods · functional analysis · limit processes.

Learning outcomes / benefits

Explain the main definitions and notation used in the chapter.
Follow the chapter's proofs without skipping hypotheses or intermediate mathematical steps.
Apply the core results to worked examples.
Recognize counterexamples and understand why theorem assumptions matter.
Use the chapter as preparation for later Real Analysis, Measure Theory, Functional Analysis or related advanced courses.

Chapter 8

Chapter 8: Uniform Convergence and Continuity

Chapter scope: Uniform limits of continuous functions, epsilon proofs, compactness, uniform continuity, Dini's theorem, sup-norm completeness and diagnostic counterexamples

What is taught

The danger
Where the proof of continuity breaks
A moving transition layer
Statement
Detailed epsilon proof
Local version
Uniform continuity
Compact domains
Motivation
Proof for an increasing sequence
Why the hypotheses matter
Continuity as an interchange of limits
Local uniform convergence is enough
Continuous functions as a metric space
Why this viewpoint matters

Definitions and foundational objects

Uniform continuity
Sup-norm metric on C(K)

Important theorems, rules and methods

Uniform-limit theorem
Key epsilon split
Three-term continuity estimate
Uniform Limit Theorem
Uniform limit of uniformly continuous functions
Heine-Cantor theorem
Uniform boundedness inherited from a bounded limit
Dini's theorem
Interchange of n-limit and point limit
Completeness of C(K)

Why this chapter is taught: Understand exactly why uniform convergence preserves continuity. Explain the main definitions and notation used in the chapter.

Applications and connections: preservation of continuity · function spaces · approximation theory.

Learning outcomes / benefits

Explain the main definitions and notation used in the chapter.
Follow the chapter's proofs without skipping hypotheses or intermediate mathematical steps.
Apply the core results to worked examples.
Recognize counterexamples and understand why theorem assumptions matter.
Use the chapter as preparation for later Real Analysis, Measure Theory, Functional Analysis or related advanced courses.

Chapter 9

Chapter 9: Series of Functions

Chapter scope: Partial sums, pointwise and uniform convergence, uniform Cauchy criterion, Weierstrass M-test, Dirichlet and Abel tests, alternating estimates, rearrangements and products

What is taught

Definition through partial sums
Absolute convergence
Necessary term condition
The criterion
Uniform tail estimate
Majorizing a functional series by a numerical series
Summation by parts
Classical trigonometric application
Continuity of the sum
Linear combinations of series
Rearrangement under absolute uniform control
Cauchy products
Uniform alternating-series estimate
Dependence on the parameter domain
Quantitative remainder design

Definitions and foundational objects

Partial sums
Series of functions
Pointwise convergence of a functional series
Uniform convergence of a functional series
Pointwise absolute convergence
Normal convergence

Important theorems, rules and methods

Weierstrass M-test
Uniform term test
Uniform Cauchy criterion for series
Uniform Dirichlet test
Uniform Abel test
Continuity of a uniformly convergent series
Cauchy product under normal convergence

Why this chapter is taught: Turn infinite sums of functions into controlled analytic objects. Explain the main definitions and notation used in the chapter.

Applications and connections: Fourier/power series · functional expansions · approximation.

Learning outcomes / benefits

Explain the main definitions and notation used in the chapter.
Follow the chapter's proofs without skipping hypotheses or intermediate mathematical steps.
Apply the core results to worked examples.
Recognize counterexamples and understand why theorem assumptions matter.
Use the chapter as preparation for later Real Analysis, Measure Theory, Functional Analysis or related advanced courses.

Chapter 10

Chapter 10: Interchanging Limits, Sums, Integrals and Derivatives

Chapter scope: Uniform convergence as the main mechanism for passing limits through continuity, Riemann integration and differentiation; termwise operations, parameter limits and sharp counterexamples

What is taught

Two operations need not commute
Uniform convergence as an error estimate
Main theorem
From partial sums to the infinite series
Error control for a truncated integrated series
Why differentiation is harder
Detailed proof
The correct theorem
Repeated differentiation
Integration may fail under pointwise convergence
Uniform convergence of functions does not control derivatives
Checklist for a suspected interchange
A parameter limit
Double limits
Decision framework

Definitions and foundational objects

Definitions are developed in the running text and prerequisite/foundation blocks; do not invent additional named definitions on the website.

Important theorems, rules and methods

Uniform integral estimate
Differentiation principle
General strategy
Uniform convergence and Riemann integration
Integral error bound
Termwise integration
Differentiation under a sequence limit
Term-by-term differentiation
Practical decision framework

Why this chapter is taught: Know when an infinite process may safely cross another operation. Explain the main definitions and notation used in the chapter.

Applications and connections: termwise calculus · analysis of infinite processes · differential equations.

Learning outcomes / benefits

Explain the main definitions and notation used in the chapter.
Follow the chapter's proofs without skipping hypotheses or intermediate mathematical steps.
Apply the core results to worked examples.
Recognize counterexamples and understand why theorem assumptions matter.
Use the chapter as preparation for later Real Analysis, Measure Theory, Functional Analysis or related advanced courses.

Chapter 11

Chapter 11: Power Series

Chapter scope: Convergence geometry, radius of convergence, Cauchy--Hadamard theory, uniform convergence on compact subintervals, endpoint analysis and termwise operations

What is taught

From numerical series to function series
Why convergence propagates inward
The root-test viewpoint
Ratio formula when a coefficient ratio exists
Uniform control inside the radius
Uniform remainder bounds
Continuity of the sum
Integration preserves the radius
Differentiation preserves the radius
The two boundary points are independent problems
Abel's theorem applied to the logarithm series
Addition, scalar multiplication and Cauchy products
Coefficients as derivatives
Preparation for Taylor theory

Definitions and foundational objects

Power series

Important theorems, rules and methods

Cauchy--Hadamard
Basic convergence geometry
Cauchy--Hadamard formula
Ratio formula
Uniform convergence on compact subintervals
Continuity inside the radius
Term-by-term integration
Term-by-term differentiation
Abel's continuity theorem at a boundary point
Uniqueness of power-series coefficients

Why this chapter is taught: Turn an infinite algebraic expansion into a rigorously controlled function. Explain the main definitions and notation used in the chapter.

Applications and connections: analytic expansions · differential equations · numerical approximation.

Learning outcomes / benefits

Explain the main definitions and notation used in the chapter.
Follow the chapter's proofs without skipping hypotheses or intermediate mathematical steps.
Apply the core results to worked examples.
Recognize counterexamples and understand why theorem assumptions matter.
Use the chapter as preparation for later Real Analysis, Measure Theory, Functional Analysis or related advanced courses.

Chapter 12

Chapter 12: Taylor and Maclaurin Series

Chapter scope: Taylor polynomials, remainder formulas, rigorous error estimates, analyticity, standard expansions and approximation by local power series

What is taught

Matching derivatives at a point
The remainder
Lagrange form of the remainder
Error estimate
Integral remainder
Infinite Taylor expansion
Analytic implies infinitely differentiable
Exponential, sine and cosine
Logarithmic and binomial expansions
Choosing the degree from a tolerance
Deriving elementary inequalities
Infinite differentiability is not enough
Why all derivatives vanish
The hierarchy of regularity

Definitions and foundational objects

Taylor polynomial
Taylor series
Real analytic function

Important theorems, rules and methods

Taylor polynomial
Lagrange remainder
Taylor's theorem with Lagrange remainder
Integral form of Taylor's remainder
Criterion for equality with the Taylor series
Classical counterexample

Why this chapter is taught: Replace a smooth function by a polynomial with quantified error. Explain the main definitions and notation used in the chapter.

Applications and connections: local approximation · error estimates · asymptotics · numerical analysis.

Learning outcomes / benefits

Explain the main definitions and notation used in the chapter.
Follow the chapter's proofs without skipping hypotheses or intermediate mathematical steps.
Apply the core results to worked examples.
Recognize counterexamples and understand why theorem assumptions matter.
Use the chapter as preparation for later Real Analysis, Measure Theory, Functional Analysis or related advanced courses.

Chapter 13

Chapter 13: Equicontinuity and Compact Families of Functions

Chapter scope: Uniform control of entire families, boundedness, equicontinuity, Arzela--Ascoli compactness and subsequence methods in spaces of continuous functions

What is taught

From one continuous function to a whole family
Boundedness of a family
Common modulus of continuity
Compactness in C([a,b])
Necessity for compact sets
Convergence on a countable dense set
Cantor diagonal selection
The finite-net estimate
Compactness from derivative estimates
Relation with existence proofs

Definitions and foundational objects

Equicontinuity
Equicontinuity at a point
Equicontinuity on a set
Uniformly bounded family
Uniform Holder family

Important theorems, rules and methods

Arzela--Ascoli idea
Equicontinuity plus boundedness at one point
Arzela--Ascoli on [a,b]
Key upgrade principle

Why this chapter is taught: Compactness for functions begins with common control in both size and oscillation. Explain the main definitions and notation used in the chapter.

Applications and connections: compactness in function spaces · existence arguments · functional analysis.

Learning outcomes / benefits

Explain the main definitions and notation used in the chapter.
Follow the chapter's proofs without skipping hypotheses or intermediate mathematical steps.
Apply the core results to worked examples.
Recognize counterexamples and understand why theorem assumptions matter.
Use the chapter as preparation for later Real Analysis, Measure Theory, Functional Analysis or related advanced courses.

Chapter 14

Chapter 14: Polynomial Approximation

Chapter scope: Weierstrass approximation, Bernstein polynomials, density in continuous-function spaces, Stone--Weierstrass theory and uniform approximation methods

What is taught

Why uniform approximation?
Polynomial density question
Approximation preserves many global properties approximately
Bernstein basis on [0,1]
Exact reproduction of constants and linear functions
Why Bernstein polynomials converge
Statement and meaning
Proof on [0,1] using Bernstein polynomials
Transfer to any compact interval
From polynomials to abstract algebras
Classical Weierstrass as a special case
Why lattice operations appear in the proof
Best approximation problem
Finite-dimensional existence
Approximation and integrals

Definitions and foundational objects

Uniform approximation error
Uniform approximation
Dense subset
Bernstein polynomial
Subalgebra of C(K)
Separating points
Best uniform approximant

Important theorems, rules and methods

Weierstrass approximation
Bernstein polynomial
Weierstrass approximation theorem
Real Stone--Weierstrass

Why this chapter is taught: Approximate every continuous profile by algebraic objects that are easy to compute. Explain the main definitions and notation used in the chapter.

Applications and connections: numerical approximation · interpolation · function algebras · approximation theory.

Learning outcomes / benefits

Explain the main definitions and notation used in the chapter.
Follow the chapter's proofs without skipping hypotheses or intermediate mathematical steps.
Apply the core results to worked examples.
Recognize counterexamples and understand why theorem assumptions matter.
Use the chapter as preparation for later Real Analysis, Measure Theory, Functional Analysis or related advanced courses.

Chapter 15

Chapter 15: Fourier Series

Chapter scope: Orthogonal trigonometric systems, Fourier coefficients, Bessel inequality, convergence theorems, half-range expansions, best mean-square approximation and Parseval theory

What is taught

Periodicity
Orthogonality
Trigonometric polynomials
Coefficient formulas
Partial sums as an integral operator
Dirichlet convergence theorem
Endpoint interpretation
Symmetry reductions
Why half-range expansions matter
Mean-square error
Bessel inequality
Parseval identity
When Fourier series converge uniformly
Gibbs phenomenon
Fejer averaging as a remedy

Definitions and foundational objects

Periodic function
Fourier coefficients
Dirichlet kernel
Half-range sine series
Half-range cosine series
Gibbs phenomenon

Important theorems, rules and methods

Fourier coefficients for period (2pi)
Dirichlet jump value
Dirichlet pointwise convergence
Best mean-square approximation
Bessel inequality
Useful sufficient criterion
Fejer's theorem (continuous case)

Why this chapter is taught: Resolve a periodic function into harmonics and understand exactly what the series converges to. Explain the main definitions and notation used in the chapter.

Applications and connections: harmonic analysis · PDE · signal processing · approximation.

Learning outcomes / benefits

Explain the main definitions and notation used in the chapter.
Follow the chapter's proofs without skipping hypotheses or intermediate mathematical steps.
Apply the core results to worked examples.
Recognize counterexamples and understand why theorem assumptions matter.
Use the chapter as preparation for later Real Analysis, Measure Theory, Functional Analysis or related advanced courses.

Chapter 16

Chapter 16: Euclidean Space and Metric Foundations

Chapter scope: Euclidean norm, distance, open and closed sets, limit points, compactness, connectedness, sequences and the Heine--Borel theorem

What is taught

From the real line to R^n
Metric-space viewpoint
Open balls
Interior points
Dense sets
Bolzano--Weierstrass in R^n

Definitions and foundational objects

Norm and metric
Euclidean inner product
Euclidean norm
Euclidean distance
Metric
Open ball
Neighborhood
Interior point and interior
Boundary point
Exterior point
Open set
Closed set
Limit point
Closure
Dense subset
Convergence in R^n
Cauchy sequence
Open cover
Compact set
Disconnected and connected sets
Path connected

Important theorems, rules and methods

Cauchy--Schwarz inequality
Triangle inequality
Basic laws of open sets
Closed-set criterion
Coordinatewise convergence
Completeness of R^n
Heine--Borel in R^n
Sequential compactness
Extreme-value principle
Path connected implies connected
Convex sets are path connected

Why this chapter is taught: Move from one-dimensional real analysis to the geometry and topology of R^n. Explain the main definitions and notation used in the chapter.

Applications and connections: metric/topological foundations · compactness · multivariable analysis.

Learning outcomes / benefits

Explain the main definitions and notation used in the chapter.
Follow the chapter's proofs without skipping hypotheses or intermediate mathematical steps.
Apply the core results to worked examples.
Recognize counterexamples and understand why theorem assumptions matter.
Use the chapter as preparation for later Real Analysis, Measure Theory, Functional Analysis or related advanced courses.

Chapter 17

Chapter 17: Limits and Continuity in Several Variables

Chapter scope: Multivariable limits, epsilon--delta and sequential criteria, path tests, continuity, compact-domain consequences and vector-valued maps

What is taught

Domain and natural domain
Graphs and level sets
Path tests
Preimage characterization
Curves as vector-valued functions

Definitions and foundational objects

Epsilon--delta limit
Scalar-valued function of several variables
Limit of a function
Continuity at a point
Vector-valued map

Important theorems, rules and methods

Uniqueness of multivariable limits
Algebra of limits
Sequential criterion for limits
Algebra and composition of continuous functions
Sequential criterion for continuity
Continuous image of a compact set
Extreme value theorem
Heine--Cantor uniform continuity
Continuous image of a connected set
Component criterion

Why this chapter is taught: Extend the language of limits and continuity from the real line to multidimensional domains. Explain the main definitions and notation used in the chapter.

Applications and connections: multivariable calculus · topology · optimization · continuity of vector-valued maps.

Learning outcomes / benefits

Explain the main definitions and notation used in the chapter.
Follow the chapter's proofs without skipping hypotheses or intermediate mathematical steps.
Apply the core results to worked examples.
Recognize counterexamples and understand why theorem assumptions matter.
Use the chapter as preparation for later Real Analysis, Measure Theory, Functional Analysis or related advanced courses.

Chapter 18

Chapter 18: Differentiability in R^n

Chapter scope: Partial and directional derivatives, total derivative, gradient, Jacobian matrix, differentiability criteria, chain rule and mean-value estimates

What is taught

Tangent-plane approximation
Partial derivatives may exist without continuity
Linearization and differentials

Definitions and foundational objects

Total derivative
Partial derivative
Directional derivative
Differentiability at a point
Gradient
Jacobian matrix

Important theorems, rules and methods

Differentiability implies continuity
Differentiability implies directional derivatives
Directional derivative via the gradient
Continuous partial derivatives imply differentiability
Multivariable chain rule
Mean-value inequality for scalar functions
Zero derivative on a connected open set

Why this chapter is taught: Learn the correct linear approximation concept behind multivariable differentiation. Explain the main definitions and notation used in the chapter.

Applications and connections: Jacobian methods · nonlinear analysis · optimization · differential geometry.

Learning outcomes / benefits

Explain the main definitions and notation used in the chapter.
Follow the chapter's proofs without skipping hypotheses or intermediate mathematical steps.
Apply the core results to worked examples.
Recognize counterexamples and understand why theorem assumptions matter.
Use the chapter as preparation for later Real Analysis, Measure Theory, Functional Analysis or related advanced courses.

Chapter 19

Chapter 19: Higher Derivatives, Taylor Theory and Extrema

Chapter scope: Second derivatives, Hessian matrix, mixed partials, multivariable Taylor theorem, unconstrained and constrained extrema, Lagrange multipliers, implicit and inverse-function bridges

What is taught

Second partial derivatives
Quadratic forms
Two-variable form
Why a constraint changes the condition
Testing candidates
Tangent-space interpretation
Local solvability
Local invertibility
Connection with constrained optimization

Definitions and foundational objects

Hessian
Hessian matrix
Positive definite matrix
Local maximum and minimum
Critical point

Important theorems, rules and methods

Two-variable test
Lagrange multipliers
Clairaut--Schwarz equality of mixed partials
Second-order Taylor expansion
First-order necessary condition
Second-derivative test in two variables
Lagrange multiplier condition
Several equality constraints
Implicit function theorem in two variables
Inverse function theorem

Why this chapter is taught: Use second-order structure to understand curvature, approximation and optimization. Explain the main definitions and notation used in the chapter.

Applications and connections: optimization · constrained extrema · implicit/inverse function ideas · local geometry.

Learning outcomes / benefits

Explain the main definitions and notation used in the chapter.
Follow the chapter's proofs without skipping hypotheses or intermediate mathematical steps.
Apply the core results to worked examples.
Recognize counterexamples and understand why theorem assumptions matter.
Use the chapter as preparation for later Real Analysis, Measure Theory, Functional Analysis or related advanced courses.

Chapter 20

Chapter 20: Multiple Riemann Integration

Chapter scope: Riemann integration on rectangles in R^n, Jordan content, iterated integrals, Fubini-type theorems, general regions, Jacobians and change of variables

What is taught

From one interval to a rectangle
Multiple Riemann sums
Darboux sums in several variables
Characteristic functions and geometric content
Integrating over a bounded region
Discontinuities and integrability
Slicing a rectangular integral
Fubini in higher dimensions
Type I and Type II planar regions
Reversing the order
Area and volume as integrals
Why the area element changes
Jacobians in R^n
Polar coordinates
Cylindrical coordinates
Spherical coordinates
A.1 What Riemann integration does very well
A.2 A basic obstruction: too many discontinuities
A.3 Limits of integrable functions
B.1 Sets of measure zero
B.2 Outer measure
B.3 Measurability
C.1 Measurable functions
C.2 Simple functions
C.3 Nonnegative measurable functions
C.4 Signed functions
C.5 A striking example
D.1 Monotone convergence
D.2 Fatou's lemma
D.3 Dominated convergence
D.4 Almost everywhere
D.5 The spaces L^p
E.1 Relationship between the two integrals
E.2 The Lebesgue criterion revisited
E.3 Which theory should be used?
E.4 Where to go next

Definitions and foundational objects

Riemann sum in higher dimensions
Closed rectangle in R^n
Mesh of a rectangular partition
Riemann integral on a rectangle
Characteristic function
Jordan measurable set
Set of Jordan content zero
Jacobian determinant
Measure-zero set in R
Lebesgue outer measure
Carathéodory measurability
Measurable real-valued function
Integral of a nonnegative simple function
Lebesgue integral of a nonnegative function
Lebesgue integrable function
Almost everywhere

Important theorems, rules and methods

Uniqueness of the multiple Riemann integral
Darboux criterion in R^n
Continuous functions are integrable
Boundary criterion for Jordan measurability
Continuous functions on Jordan regions
Lebesgue criterion for Riemann integrability - bridge statement
Fubini theorem for continuous functions on a rectangle
Integration on a vertically simple region
Change-of-variables theorem in the plane
Polar-coordinate formula
Monotone Convergence Theorem - preview
Fatou's Lemma - preview
Dominated Convergence Theorem - preview
Compatibility theorem

Why this chapter is taught: Integrate rigorously in several dimensions and connect geometry with analysis. Explain the main definitions and notation used in the chapter.

Applications and connections: multivariable integration · probability · mathematical physics · change of variables · measure theory.

Learning outcomes / benefits

Explain the main definitions and notation used in the chapter.
Follow the chapter's proofs without skipping hypotheses or intermediate mathematical steps.
Apply the core results to worked examples.
Recognize counterexamples and understand why theorem assumptions matter.
Use the chapter as preparation for later Real Analysis, Measure Theory, Functional Analysis or related advanced courses.

View PDF Download PDF Backup Copy

Advanced Bridge · From Riemann to Lebesgue

Why measure-theoretic ideas come next

Multiple Riemann integration explains what Riemann theory does well, then introduces the conceptual limitations that motivate measure and Lebesgue integration: sets of measure zero, outer measure, measurability, measurable functions, simple functions, nonnegative and signed integration, almost-everywhere reasoning, the Monotone Convergence Theorem, Fatou's Lemma, the Dominated Convergence Theorem and the spaces L^p.

This is a bridge, not a complete Measure Theory course. Continue to separate Measure Theory / Lebesgue Integration discovery.

Worked-example and solution scope

How the complete book is meant to be used

Read a definition, identify its hypotheses, work through the lemmas and theorem proof, reproduce a calculation, inspect a counterexample when one is supplied, and then attempt the nearby exercise. The HTML page summarizes problem types honestly; it does not reproduce long copyrighted textbook exercises. If a dedicated chapter PDF is not available, every chapter action remains tied to the complete owner PDF.

DefinitionsIntermediate estimatesTheorem + proofCorollaries / counterexamplesWorked examples / exercises

Programme relevance

One canonical resource across related pathways

BS Mathematics

Books, Notes & Solutions → Real Analysis → Real Analysis II.

Legacy BSc / MSc Mathematics

Classical Real Analysis, Advanced Analysis revision and course-resource discovery.

ADP, MS and MPhil

Contextual discovery, prerequisite revision or advanced undergraduate preparation where the topic overlap is genuine.

Mathematics Library

Separate owner-resource record; programme surfaces point here rather than cloning the academic body.

Curriculum / discovery context

Pakistan University / Course Crosswalk

This comparison is curriculum and discovery context only—not endorsement, official university notes or a claim that every institution uses the same outline. Course titles, codes, semester placement and topic boundaries vary.

HEC Pakistan — National Mathematics Curriculum 2025

REAL ANALYSIS-II outcomes include tests for convergence of sequences of functions, Riemann-Stieltjes integrals, and proper/improper integrals.

Open supplied course context ↗

University of the Punjab

MATH-307 Real Analysis-II, 3 credit hours: Riemann-Stieltjes integration, bounded variation, improper integrals, Beta/Gamma, sequences/series of functions, power series, pointwise/uniform convergence, and preservation of continuity/integration/differentiation.

Open supplied course context ↗

University of Sargodha

MATH-6105 Real Analysis-II, 3(3-0): Riemann-Stieltjes integrals, bounded variation, improper integrals, Beta/Gamma, sequences and series of functions, pointwise/uniform convergence and related calculus results.

Open supplied course context ↗

COMSATS University Islamabad — Attock course-material evidence

MTH322 Real Analysis-II course materials cover sequences/series of functions, uniform convergence, Weierstrass M-test, power series, termwise operations and improper integrals.

Verify current course information from the institution.

Virtual University of Pakistan

MTH631 Real Analysis II, undergraduate 3 credit hours: uniform convergence, power series, equicontinuity, Stone-Weierstrass, Fourier series, Beta/Gamma, functions of several variables, R^n differentiability, extrema, improper/multiple integrals and bounded variation.

Open supplied course context ↗

LUMS

MATH 309 Introduction to Analysis II, 3 credit hours: open/closed/compact sets, sequences and series of functions, pointwise/uniform convergence, power/Taylor series and basic metric spaces.

Open supplied course context ↗

University of Gujrat

MATH-302 Real Analysis-II, Semester VI, 3 credit hours in the BS Mathematics scheme.

Open supplied course context ↗

University of Management and Technology (UMT)

MA-314 Real Analysis-II, 3 credit hours in the BS Mathematics / ADP lateral-entry roadmap.

Open supplied course context ↗

Lahore College for Women University (LCWU)

MATH-351 Real Analysis-II, 3 credit hours in Semester VI.

Open supplied course context ↗

The University of Lahore

Faculty/course evidence lists Real Analysis-II in Semester VI, with Measure Theory & Lebesgue Integration later in the programme.

Open supplied course context ↗

Wider institutional discovery order

Major and public institutions first

This is a curated The Math Hub presentation order for discovery consistency, not an objective national ranking. Only the individually described crosswalk entries above carry specific course-context wording.

Show public / major discovery order (31)

University of the Punjab
National University of Sciences & Technology (NUST)
Quaid-i-Azam University
COMSATS University Islamabad
Virtual University of Pakistan
University of Karachi
International Islamic University Islamabad (IIUI)
University of Peshawar
Government College University Faisalabad (GCUF)
Bahauddin Zakariya University (BZU)
University of Sargodha (UOS)
Government College University Lahore
University of Gujrat
Lahore College for Women University (LCWU)
Abdul Wali Khan University Mardan (AWKUM)
BUITEMS
The Islamia University of Bahawalpur
University of Narowal
University of Swat
University of Malakand
University of Balochistan
Gomal University
Hazara University
Karakoram International University
University of Azad Jammu & Kashmir
University of Education Lahore
University of Okara
University of Jhang
Baba Guru Nanak University
University of Mianwali
University of Sahiwal

Show private / non-public discovery order (12)

LUMS
The University of Lahore
University of Management and Technology (UMT)
University of Central Punjab (UCP)
Forman Christian College (FCCU)
Superior University
Minhaj University Lahore
Riphah International University
Habib University
Greenwich University
University of South Asia
University of Lahore — Sargodha Campus

Formal / lawful references

Recommended Reference Books

These books are alignment and further-study references. No copyrighted reference-book text is reproduced here.

Walter Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill ↗

Riemann-Stieltjes integration, sequences/series of functions, special functions and functions of several variables.

Robert G. Bartle & Donald R. Sherbert, Introduction to Real Analysis, 4th ed., Wiley ↗

Student-friendly rigorous analysis reference; also aligns with University of Sargodha recommended texts.

Tom M. Apostol, Mathematical Analysis: A Modern Approach to Advanced Calculus, 2nd ed., Pearson ↗

Bounded variation, Riemann-Stieltjes integration, sequences of functions, Fourier series, multivariable differential calculus and multiple Riemann integrals.

S. Ponnusamy, Foundations of Mathematical Analysis, 1st ed., Birkhäuser / Springer ↗

Improper integrals, power series, uniform convergence, Fourier series, bounded variation and Riemann-Stieltjes integrals.

William F. Trench, Introduction to Real Analysis, Open edition, Trinity University ↗

Legally free open-access textbook with supplements including functions defined by improper integrals and Lagrange multipliers.

Witold A. J. Kosmala, A Friendly Introduction to Analysis, 2nd ed., Pearson ↗

Readable undergraduate reference covering sequences/series of functions, vector calculus, functions of two variables and multiple integration.

S. C. Malik & Savita Arora, Mathematical Analysis, New Age International ↗

Classical undergraduate/postgraduate style with Riemann-Stieltjes, improper integrals, uniform convergence, power/Fourier series, several variables and metric spaces.

Jiří Lebl, Basic Analysis II: Introduction to Real Analysis, Volume II, Version 6.3 (2026) ↗

Open textbook for multivariable analysis, multivariable Riemann integration, power series, Arzelà-Ascoli, Stone-Weierstrass and Fourier series.

Brian S. Thomson, Judith B. Bruckner & Andrew M. Bruckner, Elementary Real Analysis, 2nd ed. ↗

Two-semester undergraduate real analysis with Volume II devoted to functions, power series, Euclidean spaces and differentiation on R^n.

Free / legal further resources

William F. Trench — Introduction to Real Analysis (open) ↗Jiří Lebl — Basic Analysis I & II (open/OER) ↗Elementary Real Analysis — Open Textbook Initiative ↗Virtual University MTH631 course overview ↗The Math Hub Mathematics Library — Real Analysis search/discovery →

Academic acknowledgement

Prof. Syed Gul Shah

The authors gratefully acknowledge the formative academic influence of Prof. Syed Gul Shah, former Chairman of the Department of Mathematics, University of Sargodha. His teaching and detailed Real Analysis notes helped generations of students approach rigorous analysis through clear, carefully developed mathematical steps. This acknowledgement reflects academic gratitude and mentorship; it does not imply co-authorship or institutional endorsement.

Student questions · 80 answers

Real Analysis II FAQs

These visible FAQs support student understanding and internal search. They do not promise Google FAQ rich results.

What is Real Analysis II?

Real Analysis II is a proof-based continuation of Real Analysis I. In this resource it develops rigorous integration, Riemann-Stieltjes theory, bounded variation, improper integrals, sequences and series of functions, uniform convergence, power/Fourier series and analysis in R^n.

Is Real Analysis II the same at every university?

No. Universities use overlapping but not identical outlines. The Math Hub resource therefore keeps the HEC/Pakistan core visible while also covering broader BS and classical BSc/MSc analysis topics used across different institutions.

What should I know before starting Real Analysis II?

A learner should know basic algebra, functions, limits, continuity, differentiation, elementary integration, sequences and series, and basic proof language. The book is deliberately written from foundations toward advanced topics so missing intermediate steps are minimized.

Does this book cover Riemann-Stieltjes integration?

Yes. It treats the definition, Stieltjes-Darboux sums, existence criteria, refinement, properties, step integrators, integration by parts, change of variables and the relationship with ordinary Riemann integration.

Does this book cover functions of bounded variation?

Yes. It develops total variation, monotone functions, algebra of BV functions, Jordan-type decomposition ideas, Stieltjes connections and rectifiable-curve/arc-length connections.

Are improper integrals included?

Yes. The resource covers infinite intervals, finite singularities, mixed improper integrals, convergence tests, absolute/conditional convergence, parameter-dependent improper integrals and uniform convergence issues.

Are Beta and Gamma functions included?

Yes. Definitions, convergence, recurrence identities, factorial extension, half-integer values, the Beta-Gamma relationship and related integral calculations are included.

Does the book explain pointwise and uniform convergence?

Yes. It develops both definitions from the quantifiers, gives examples and counterexamples, presents supremum/Cauchy criteria, and studies what uniform convergence preserves.

Does it include power series and Taylor series?

Yes. Radius of convergence, Cauchy-Hadamard ideas, endpoint behavior, termwise differentiation/integration, Taylor polynomials, remainder forms, Maclaurin series and analytic-function issues are covered.

Are Fourier series included?

Yes. Fourier coefficients, trigonometric series, convergence criteria, symmetry, approximation, Bessel/Parseval-type results and further Fourier-series ideas are treated.

Does it cover multivariable Real Analysis?

Yes. Euclidean space, norms, compactness, functions of several variables, multivariable limits/continuity, differentiability in R^n, gradients, Jacobians, Hessians, extrema and multiple Riemann integration are included.

Is Measure Theory fully covered?

No. The book ends with an advanced bridge from Riemann integration toward Lebesgue/measure-theoretic ideas. A full Measure Theory course should remain a separate resource.

Who prepared the Real Analysis II resource?

The resource is prepared by Mehreen Kanwal (MPhil Mathematics) and Rana Ali Hasan (MPhil Mathematics), with both names linked to their canonical The Math Hub educator profiles.

Who is Prof. Syed Gul Shah in relation to this resource?

Prof. Syed Gul Shah, former Chairman of the Department of Mathematics at the University of Sargodha, is acknowledged for his formative teaching influence and detailed Real Analysis notes. He is not presented as a co-author and the acknowledgement does not imply university endorsement.

Are Prof. Syed Gul Shah's notes listed as a formal reference book?

No. The public acknowledgement may mention his academic influence, while the formal Recommended Reference Books section should contain established books and lawful/official links.

Can I view the PDF online?

Yes. The View PDF action opens the verified R2 PDF in a new tab.

Can I download the PDF?

Yes. A separate Download PDF action uses the same verified owner-controlled R2 source and the site's established download behavior.

Is there a backup copy?

Yes. The Backup Copy action opens the verified Google Drive copy. The visible button label remains 'Backup Copy' rather than 'Google Drive'.

Is the resource official university material?

No. The Math Hub is an independent educational resource. University names are used only for curriculum/discovery context and must never be described as endorsements or official notes.

Why are many university outlines compared?

Cross-university comparison helps students see which topics are common, which are extended, and how a broad Real Analysis II resource can support learners from different Pakistani programmes without creating duplicate university-specific pages.

What is studied in Chapter 1: Riemann Integration: Rigorous Foundations?

This chapter develops Why integration needs partitions, Refinement, Local upper and lower bounds, Geometric interpretation. It progresses from definitions and basic calculations to theorem-level reasoning, worked applications and practice.

Which important results appear in Riemann Integration: Rigorous Foundations?

Representative results and methods include Refinement improves Darboux bounds, Every lower sum is below every upper sum, Fundamental inequality. Only results genuinely present in the current book should be named on the public page.

How should a student study Riemann Integration: Rigorous Foundations?