# Advanced Analysis: Handwritten Notes

These notes are provided by Mr. Anwar Khan. We are really very thankful to Mr. Anwar Khan for providing these notes and appreciates his effort to publish these notes on MathCity.org

It covers the complete syllabus of Advanced Analysis paper of MSc Mathematics. See the contents of the notes given below to see the topics covered by these notes.

Name | Advanced Analysis: Handwritten Notes |
---|---|

Provider | Mr. Anwar Khan |

Pages | 263 pages |

Format | PDF (see Software section for PDF Reader) |

Size | 7.5 MB |

### CONTENTS OR SUMMARY:

- Equivalent Set
- Symmetric property
- Transitive Property
- Finite Set
- Infinite Set
- Denumerable Set
- Countable Set
- Geometrical Interpretation of Equivalent Set
- Duality
- Choice Function
- Characteristic Function
- Cardinal Numbers
- Arithmetic Cardinal Number
- Transfinite Cardinal Numbers
- Ordering of Cardinal Number
- Exponent of Cardinal Numbers
- Ordered Set
- Partially Ordered Set
- Reflexive
- Anti-symmetric
- Transitive
- Inverse Ordered
- First Element
- Last Element
- Maximal Element
- Minimal Element
- The Linearly Ordered Set
- Lower Bound
- Upper Bound
- Infimum
- Suprimum
- Similar Sets
- Order type (Ordinal Number)
- Well Ordered Set
- Principal of Mathematical Induction
- Initial Segment
- Immediate Successor & Immediate Predecessor
- Shorter & Longer Sets
- Ordinal Number
- Transfinite Ordinal Number

- Inequality and Ordinal Numbers
- Ordinal Addition
- Ordinal Multiplication
- Structure of Ordinal Number
- Ordinary Differential Equation
- Hyper-geometric Series
- Gamma Function
- Beta Function
- Relation between Gama and Beta Function
- Bessel Function
- Differential Recurrence relation of Bessel Function
- Spherical Bessel Function
- Properties of Bessel Function
- Hyper-geometric Function
- Legender Polynomial
- Generating Function
- Bessel’s differential Equation
- Measure Theory
- Set Function
- Non-negative Function
- Additivity of a Set Function
- Finite Additive Set Function
- Sub- additivity of a Set Function
- Finitely sub Additive of a Set Function
- Countable sub additive of a Set Function
- Pre-measure
- Outer Measure
- Monotone Set Function
- Scalar Multiplication of an Outer Measure
- Lebegue Outer Measure of a Set of Real Number
- Translation of a Set
- Monotone Property
- Lebesgue Measure of a set of Real Number
- Boolean Algebra or Simply Algebra of Sets
- Sigma Algebra
- Boral Subset of Real Line
- Lebesgue Measure of Measurable Set
- Measurable Function
- Almost Everywhere
- Boral Measurable

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