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Notes of Principles of Mathematical Analysis

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# POMA Notes

Study notes of _Principles of Mathematical Analysis_.

This project is written in Chinese and is under long-term construction, check [PDF](./out/poma-notes.pdf) for the latest output.

## Catalog

1. [The Real and Complex Number Systems](./Ch01%20The%20Real%20and%20Complex%20Number%20Systems/The%20Real%20and%20Complex%20Number%20Systems.tex)

1. Basic Topology

1. [Finite, Countable, and Uncountable Sets](./Ch02%20Basic%20Topology/Sec01%20Finite%2C%20Countable%2C%20and%20Uncountable%20Sets.tex)

1. [Compact Sets](./Ch02%20Basic%20Topology/Sec02%20Compact%20Sets.tex)

1. [Perfect Sets](./Ch02%20Basic%20Topology/Sec03%20Perfect%20Sets.tex)

1. [Connected Sets](./Ch02%20Basic%20Topology/Sec04%20Connected%20Sets.tex)

1. Numerical Sequences and Series

1. [Convergent Sequences](./Ch03%20Numerical%20Sequences%20and%20Series/Sec01%20Convergent%20Sequences.tex)

1. [Subsequences](./Ch03%20Numerical%20Sequences%20and%20Series/Sec02%20Subsequences.tex)

1. [Cauchy Sequences](./Ch03%20Numerical%20Sequences%20and%20Series/Sec03%20Cauchy%20Sequences.tex)

1. [Upper and Lower Limits](./Ch03%20Numerical%20Sequences%20and%20Series/Sec04%20Upper%20and%20Lower%20Limits.tex)

1. [Some Special Sequences](./Ch03%20Numerical%20Sequences%20and%20Series/Sec05%20Some%20Special%20Sequences.tex)

1. [Series](./Ch03%20Numerical%20Sequences%20and%20Series/Sec06%20Series.tex)

1. [Series of Nonnegative Terms](./Ch03%20Numerical%20Sequences%20and%20Series/Sec07%20Series%20of%20Nonnegative%20Terms.tex)

1. [The Number e](./Ch03%20Numerical%20Sequences%20and%20Series/Sec08%20The%20Number%20e.tex)

1. [The Root and Ratio Tests](./Ch03%20Numerical%20Sequences%20and%20Series/Sec09%20The%20Root%20and%20Ratio%20Tests.tex)

1. [Power Series](./Ch03%20Numerical%20Sequences%20and%20Series/Sec10%20Power%20Series.tex)

1. [Summation by Parts](./Ch03%20Numerical%20Sequences%20and%20Series/Sec11%20Summation%20by%20Parts.tex)

1. [Absolute Convergence](./Ch03%20Numerical%20Sequences%20and%20Series/Sec12%20Absolute%20Convergence.tex)

1. [Addition and Multiplication of Series](./Ch03%20Numerical%20Sequences%20and%20Series/Sec13%20Addition%20and%20Multiplication%20of%20Series.tex)

1. [Rearrangements](./Ch03%20Numerical%20Sequences%20and%20Series/Sec14%20Rearrangements.tex)

1. Continuity

1. [Limits of Functions](./Ch04%20Continuity/Sec01%20Limits%20of%20Functions.tex)

1. [Continuous Functions](./Ch04%20Continuity/Sec02%20Continuous%20Functions.tex)

1. [Continuity and Compactness](./Ch04%20Continuity/Sec03%20Continuity%20and%20Compactness.tex)

1. [Continuity and Connectedness](./Ch04%20Continuity/Sec04%20Continuity%20and%20Connectedness.tex)

1. [Discontinuities](./Ch04%20Continuity/Sec05%20Discontinuities.tex)

1. [Monotonic Functions](./Ch04%20Continuity/Sec06%20Monotonic%20Functions.tex)

1. [Infinite Limits and Limits at Infinity](./Ch04%20Continuity/Sec07%20Infinite%20Limits%20and%20Limits%20at%20Infinity.tex)

1. Differentiation

1. [The Derivative of a Real Function](./Ch05%20Differentiation/Sec01%20The%20Derivative%20of%20a%20Real%20Function.tex)

1. [Mean Value Theorems](/Ch05%20Differentiation/Sec02%20Mean%20Value%20Theorems.tex)

1. [The Continuity of Derivatives](./Ch05%20Differentiation/Sec03%20The%20Continuity%20of%20Derivatives.tex)

1. [L'Hospital's Rule](./Ch05%20Differentiation/Sec04%20L'Hospital's%20Rule.tex)

1. [Derivatives of Higher Order](./Ch05%20Differentiation/Sec05%20Derivatives%20of%20Higher%20Order.tex)

1. [Taylor's Theorem](./Ch05%20Differentiation/Sec06%20Taylor's%20Theorem.tex)

1. [Differentiation of Vector-valued Functions](./Ch05%20Differentiation/Sec07%20Differentiation%20of%20Vector-valued%20Functions.tex)

1. The Riemann-Stieltjes Integral

1. [Definition and Existence of the Integral](./Ch06%20The%20Riemann-Stieltjes%20Integral/Sec01%20Definition%20and%20Existence%20of%20the%20Integral.tex)

<!-- 1. [Properties of the Integral](./Ch06%20The%20Riemann-Stieltjes%20Integral/Sec02%20Properties%20of%20the%20Integral.tex)

1. [Integration and Differentiation](./Ch06%20The%20Riemann-Stieltjes%20Integral/Sec03%20Integration%20and%20Differentiation.tex)

1. [Integration of Vector-valued Functions](./Ch06%20The%20Riemann-Stieltjes%20Integral/Sec04%20Integration%20of%20Vector-valued%20Functions.tex)

1. [Rectifiable Curves](./Ch06%20The%20Riemann-Stieltjes%20Integral/Sec05%20Rectifiable%20Curves.tex) -->

1. Sequences and Series of Functions

<!-- 1. [Discussion of Main Problem](./Ch07%20Sequences%20and%20Series%20of%20Functions/Sec01%20Discussion%20of%20Main%20Problem.tex)

1. [Uniform Convergence](./Ch07%20Sequences%20and%20Series%20of%20Functions/Sec02%20Uniform%20Convergence.tex)

1. [Uniform Convergence and Continuity](./Ch07%20Sequences%20and%20Series%20of%20Functions/Sec03%20Uniform%20Convergence%20and%20Continuity.tex)

1. [Uniform Convergence and Integration](./Ch07%20Sequences%20and%20Series%20of%20Functions/Sec04%20Uniform%20Convergence%20and%20Integration.tex)

1. [Uniform Convergence and Differentiation](./Ch07%20Sequences%20and%20Series%20of%20Functions/Sec05%20Uniform%20Convergence%20and%20Differentiation.tex)

1. [Equicontinuous Families of Functions](./Ch07%20Sequences%20and%20Series%20of%20Functions/Sec06%20Equicontinuous%20Families%20of%20Functions.tex)

1. [The Stone-Weierstrass Theorem](./Ch07%20Sequences%20and%20Series%20of%20Functions/Sec07%20The%20Stone-Weierstrass%20Theorem.tex) -->

1. Some Special Functions

<!-- 1. [Power Series](./Ch08%20Some%20Special%20Functions/Sec01%20Power%20Series.tex)

1. [The Exponential and Logarithmic Functions](./Ch08%20Some%20Special%20Functions/Sec02%20The%20Exponential%20and%20Logarithmic%20Functions.tex)

1. [The Trigonometric Functions](./Ch08%20Some%20Special%20Functions/Sec03%20The%20Trigonometric%20Functions.tex)

1. [The Algebraic Completeness of the Complex Field](./Ch08%20Some%20Special%20Functions/Sec04%20The%20Algebraic%20Completeness%20of%20the%20Complex%20Field.tex)

1. [Fourier Series](./Ch08%20Some%20Special%20Functions/Sec05%20Fourier%20Series.tex)

1. [The Gamma Function](./Ch08%20Some%20Special%20Functions/Sec06%20The%20Gamma%20Function.tex) -->

1. Functions of Several Variables

<!-- 1. [Linear Transformations](./Ch09%20Functions%20of%20Several%20Variables/Sec01%20Linear%20Transformations.tex)

1. [Differentiation](./Ch09%20Functions%20of%20Several%20Variables/Sec02%20Differentiation.tex)

1. [The Contraction Principle](./Ch09%20Functions%20of%20Several%20Variables/Sec03%20The%20Contraction%20Principle.tex)

1. [The Inverse Function Theorem](./Ch09%20Functions%20of%20Several%20Variables/Sec04%20The%20Inverse%20Function%20Theorem.tex)

1. [The Implicit Function Theorem](./Ch09%20Functions%20of%20Several%20Variables/Sec05%20The%20Implicit%20Function%20Theorem.tex)

1. [The Rank Theorem](./Ch09%20Functions%20of%20Several%20Variables/Sec06%20The%20Rank%20Theorem.tex)

1. [Determinants](./Ch09%20Functions%20of%20Several%20Variables/Sec07%20Determinants.tex)

1. [Derivatives of Higher Order](./Ch09%20Functions%20of%20Several%20Variables/Sec08%20Derivatives%20of%20Higher%20Order.tex)

1. [Differentiation of Integrals](./Ch09%20Functions%20of%20Several%20Variables/Sec09%20Differentiation%20of%20Integrals.tex); -->

1. Integration of Differential Forms

<!-- 1. [Integration](./Ch10%20Integration%20of%20Differential%20Forms/Sec01%20Integration.tex)

1. [Primitive Mappings](./Ch10%20Integration%20of%20Differential%20Forms/Sec02%20Primitive%20Mappings.tex)

1. [Partitions of Unity](./Ch10%20Integration%20of%20Differential%20Forms/Sec03%20Partitions%20of%20Unity.tex)

1. [Change of Variables](./Ch10%20Integration%20of%20Differential%20Forms/Sec04%20Change%20of%20Variables.tex)

1. [Differential Forms](./Ch10%20Integration%20of%20Differential%20Forms/Sec05%20Differential%20Forms.tex)

1. [Simplexes and Chains](./Ch10%20Integration%20of%20Differential%20Forms/Sec06%20Simplexes%20and%20Chains.tex)

1. [Stokes' Theorem](./Ch10%20Integration%20of%20Differential%20Forms/Sec07%20Stokes'%20Theorem.tex)

1. [Closed Forms and Exact Forms](./Ch10%20Integration%20of%20Differential%20Forms/Sec08%20Closed%20Forms%20and%20Exact%20Forms.tex)

1. [Vector Analysis](./Ch10%20Integration%20of%20Differential%20Forms/Sec09%20Vector%20Analysis.tex) -->

1. The Lebesgue Theory

<!-- 1. [Set Functions](./Ch11%20The%20Lebesgue%20Theory/Sec01%20Set%20Functions.tex)

1. [Construction of the Lebesgue Measure](./Ch11%20The%20Lebesgue%20Theory/Sec02%20Construction%20of%20the%20Lebesgue%20Measure.tex)

1. [Measure Spaces](./Ch11%20The%20Lebesgue%20Theory/Sec03%20Measure%20Spaces.tex)

1. [Measurable Functions](./Ch11%20The%20Lebesgue%20Theory/Sec04%20Measurable%20Functions.tex)

1. [Simple Functions](./Ch11%20The%20Lebesgue%20Theory/Sec05%20Simple%20Functions.tex)

1. [Integration](./Ch11%20The%20Lebesgue%20Theory/Sec06%20Integration.tex)

1. [Comparison with the Riemann Integral](./Ch11%20The%20Lebesgue%20Theory/Sec07%20Comparison%20with%20the%20Riemann%20Integral.tex)

1. [Integration of Complex Functions](./Ch11%20The%20Lebesgue%20Theory/Sec08%20Integration%20of%20Complex%20Functions.tex)

1. [Functions of Class L2](./Ch11%20The%20Lebesgue%20Theory/Sec09%20Functions%20of%20Class%20L2.tex) -->

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