# Mathematics

This is my attempt at digitalizing (and translating) my math notes from uni. 
It contains all the maths I learned in four semesters of university, however I have not proofread it yet, and I also plan on adding in all the proofs that were labeled as "left to the lecture attendant".

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
   7.6 Lebesgue-Stieltjes Integral

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 Outlook: Tempered Distributions

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

13. Spectral Theory  
   13.1 Spectral Theory of Bounded Linear Operators  
   13.2 Spectral Representation of Bounded Self-Adjoint Operators
   13.3 Compact & Unbounded Linear Operators