Introduction to Analysis-III

Metric & its Examples

Some examples of Matric spaces

Open Sphere

open set in metric space

Some Theorem on Open set in Metric space

Interior of set

Interior of set and Limit point

Closed Set

Closure of Set & Boundary of set

Cantor is closed

Properties of Closure

Properties of Closure of set

Dense set

Convergent Sequence, Cauchy Sequence and complete metric space

Limit point & limit of sequence

Cantor Intersection Theorem

Baire Theorem

Limit of function

Continuous function

f is continuous if and only if Inverse Image of open set is open

f is continuous if and only if inverse image of closed set is closed

Homeomorphic Metric spaces

Totally bounded set

Total bounded set and bounded set in R

Compact Metric Space

Heine-Boral Properties

Finite intersection Property and Compact Metric space

Continuous function on Compact metric space

Counter Examples on Continuous image of Cauchy Seq. and complete set

Real valued Continuous function on Compact set

Uniform continuous and compact Set

Open set in metric Subspace

Connected metric space

pointwise Convergence

pointwise convergence counter example

Uniform convergence sequence

Examples on uniform convergence sequence

Uniform convergence series and Weierstrass M-test

Examples on uniform convergence of Series

Continuity under Uniform convergence

continuous everywhere but differentiable nowhere

Change of integration and summation under uniform convergence

Uniform convergence and derivative

Term by term differentiation of Series

Uniform convergence of Power series

Identity Theorem for Power series

Abel's Limit Theorem

Cauchy Product application of Abel's Limit Theorem

Application of Abel's Limit Theorem

Stirling Formula

Bernstein polynomial

WEIERSTRASS APPROXIMATION THEOREM

Existence and Uniqueness of Exponential function

The Logarithm Function

Existence and Uniqueness of Trigonometric functions