Chapter 1 Distance Space

1.1 Definitions and Examples
1.2 General topology in metric space
1.3 Relative phase
1.4 Cauchy sequences and complete metric spaces
1.5 Compact metric space

Chapter 2 Continuous functions in metric space

2.1 Continuous functions
2.2 Continuity and product space
2.3 Continuity and Compactness
2.4 Continuity and Connectivity
2.5 Topological space

Chapter 3 Uniform Convergence

3.1 Limits of functions
3.2 Pointwise convergence and uniform convergence
3.3 Uniform convergence and continuity
3.4 Uniform convergence distance
3.5 Function term series: Weierstrass M-criteria
3.6 Uniform convergence and integration
3.7 Uniform convergence and differentiation
3.8 Uniform approximations of polynomials

Chapter 4 Power Series

4.1 Formal power series
4.2 Real analysis function
4.3 Abel's theorem
4.4 Power series product
4.5 Exponential and Logarithmic Functions
4.6 A side note about complex numbers
4.7 Trigonometric functions

Chapter 5 Fourier Series

5.1 Periodic functions
5.2 Inner product of periodic functions
5.3 Trigonometric polynomials
5.4 Periodic convolution
5.5 Fourier's theorem and Plancherel's formula

Chapter 6 Multivariable Calculus

6.1 Linear transformation
6.2 Derivatives in multivariable calculus
6.3 Partial and directional derivatives
6.4 Chain Rule in Multivariable Calculus
6.5 Second-Order Derivatives and Clairaut's Theorem
6.6 Summary of abbreviated ideas
6.7 Inverse Function Theorem in Multivariable Differential and Integral Calculus
6.8 Implicit Function Theorem

Chapter 7 Lebesgue Measure

7.1 Purpose: Lebesgue measure
7.2 First attempt: External view
7.3 The outer dimension does not have any legality.
7.4 Measurable sets
7.5 Measurable functions

Chapter 8 Lebesgue Integral

8.1 Simple functions
8.2 Integral of nonnegative measurable functions
8.3 Integration of absolutely integrable functions
8.4 Comparison with Riemann integral
8.5 Pubini's theorem

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