Table of Contents

**MTH631: Real Analysis II**

In mathematics, real analysis is a branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real functions. MTH631 Handouts pdf

# MTH631 Handouts pdf

**Course Category: Mathematics**

**Course Outline**

Sequence of functions, pointwise convergence, Uniform convergence, Theorem 4.4.4 Necessary and Sufficient conditions for pointwise and uniform convergence, Uniform convergence implies pointwise convergence, Uniform Convergence some conclusions, Integrability of the uniform limit, Uniform convergence of derivatives of a sequence of functions, Infinite Series of functions, Cauchy’s Uniform Convergence Criterion, Dominated series of real numbers for a series of functions, Weierstrass’s Test, Dirichlet’s test for uniform convergence, Series of the product of two functions, Continuity of uniformly convergent series of functions, Interchange of Summation and Integration, Integration of sequences of integrable functions, Interchange of summation and integration,

Differentiation of a sequence of functions, Interchange of summation and differentiation of infinite series, Properties of functions defined by power series, kth order derivative of a power series, uniqueness of the power series, the definite integral of a function represented by power series, Arithmetic operations with power series, a product of two functions represented by power series, the reciprocal of power series and example, Abel’s Theorem, Equicontinuous functions on a set, Uniformly convergent sequence of functions is equicontinuous, The Stone-Weierstrass Theorem, Fourier Series, Periodic functions, Trigonometric Polynomials, The space E and inner product Lemma, Orthonormal set of functions, complete set of functions, Fourier coefficients,

Even and odd functions, Convergence of Fourier Series, Dirichlet Theorem, Best Approximation Theorem, The Euler Gama functions Theorem, Convex function, The beta function, Functions of several variables, the structure of R^n, Inner product and Schwarz’s inequality in R^n, Line segments in R^n, Neighbourhoods and open sets in R^n, Cauchy’s Convergence Criterion, Principle of Nested Sets Theorem, Heine-Borel Theorem in R^n Theorem, Connected sets and Regions in R^n, Polygonally connected set in R^n, Limit of real-valued functions of n variables in R^n, Algebra of limits, infinite limits and limits at infinity of function with n variables, Vector-Valued Functions, Composite vector valued Functions and limits of vector-valued functions, Bounded functions,

Intermediate value Theorem in R^n, Uniform continuity, Directional Derivative, Differentiable Functions of Several Variables, The differential in one variable, The differential in functions of several variables, maxima and minima for functions of n variables, Differentiability of vector-valued functions, Higher derivatives of Composite functions, Higher Differentials, Vector-valued functions using matrices, Linear transformations, A New Notation for the Differential, The Norm of a Matrix, Square matrix, Continuous Transformations Theorem, Differentiable Transformations Theorem, Local invertibility of linear transformation, The implicit function theorem, Jacobians, Locally integrable functions, Absolute integrability, Conditional convergence of improper integrals,

The Dirichlet’s Test Theorem, Riemann sum in R^n, Upper and Lower Integrals, Sets with zero content, Integrals over more general subsets of R^n, Differentiable surfaces, Integrated integrals, Fubini’s Theorem, Functions of bounded variations, Additive property of total variation. MTH631 Handouts pdf

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### MTH631 HANDOUTS

**MTH631: Real Analysis II**

Some specific properties of real sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability, and integrability.

The real analysis differs from complex analysis, which deals with the study of complex numbers and their functions.

Real analysis theorems rely on the properties of the real number system to be established. The real number system consists of an uncountable set ( ), together with two binary operations denoted + and ⋅ and an order denoted <. The operations turn the real numbers into an array and, together with the order, into an ordered array. A system of real numbers is a unique complete ordered field in the sense that any other complete ordered field is isomorphic to it.