Mathematics

This is my attempt at digitalizing (and translating) my math notes from uni. It's not finished yet, I'll update it bit by bit when I feel like it

The topics covered in this script will be:

1. Fundamentals and Notation
   1.1 Logic
   1.2 Sets and Functions
   1.3 Numbers

2. Analysis: Part 1
   2.1 Elementary Inequalities
   2.2 Sequences and Limits
   2.3 Convergence of Series

3. Linear Algebra
   3.1 Vector spaces
   3.2 Matrices and Gaussian elimination
   3.3 The Determinant
   3.4 Scalar Product
   3.5 Eigenvalue problems

4. Analysis: Part 2
   4.1 Limits of Functions
   4.2 Differential Calculus

5. Topology in Metric spaces
   5.1 Metric and Normed spaces
   5.2 Sequences, Series and Limits
   5.3 Open and Closed Sets
   5.4 Continuiuty
   5.5 Convergence of Function Sequences

6. Differential Calculus for Functions with multiple Variables
   6.1 Partial and Total Differentiability
   6.2 Higher Derivatives
   6.3 Function Sequences and Differentiability
   6.4 The Banach Fixed-Point Theorem and the Implicit Function Theorem

7. Measures and Integrals
   7.1 Contents and Measures
   7.2 Integrals
   7.3 Integrals over the real numbers
   7.4 Product Measures and the Fubini Theorem
   7.5 The Transformation Theorem

8. Ordinary Differential Equations
   8.1 Solution Methods
   8.2 The Picard-Lindelöf Theorem
   8.3 Linear Differential Equation Systems

9. Integration over Submanifolds
   9.1 Line Integrals
   9.2 Surface Integrals
   9.3 Ingegral Theorems

10. Elements of Complex Analysis
    10.1 Complex Differentiability
    10.2 Complex Line Integrals
    10.3 Identity Theorems and Analytic Continuation
    10.4 Laurent Series
    10.5 Residue Theorem
    10.6 Application: Potential Theory

11. Fourier Transform and Basics of Distribution Theory
    11.1 Fourier Transform on L¹(ℝᵈ)
    11.2 Fourier Transform on L²(ℝᵈ)
    11.3 Tempered Distributions

12. Operator Theory
    12.1 Linear Operators
    12.2 Dual Spaces
    12.3 Hilbert Spaces
    12.4 Orthonormal Sets
    12.5 Adjoint Operators

13. Spectral Theory
    13.1 Spectral Theory of Bounded Linear Operators
    13.2 Spectral Representation of Bounded Self-Adjoint Operators I
    13.3 Spectral Representation of Bounded Self-Adjoint Operators II
    13.4 Compact Linear Operators
    13.5 Unbounded Linear Operators
    13.6 Spectral Representation of Unbounded Self-Adjoint Operaotrs

14. Curves in ℝ³

15. Differentiable Manifolds