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:
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Fundamentals and Notation
1.1 Logic
1.2 Sets and Functions
1.3 Numbers -
Analysis: Part 1
2.1 Elementary Inequalities
2.2 Sequences and Limits
2.3 Convergence of Series -
Linear Algebra
3.1 Vector spaces
3.2 Matrices and Gaussian elimination
3.3 The Determinant
3.4 Scalar Product
3.5 Eigenvalue problems -
Analysis: Part 2
4.1 Limits of Functions
4.2 Differential Calculus -
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 -
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 -
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 -
Ordinary Differential Equations
8.1 Solution Methods
8.2 The Picard-Lindelöf Theorem
8.3 Linear Differential Equation Systems -
Integration over Submanifolds
9.1 Line Integrals
9.2 Surface Integrals
9.3 Ingegral Theorems -
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 -
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 -
Operator Theory
12.1 Linear Operators
12.2 Dual Spaces
12.3 Hilbert Spaces
12.4 Adjoint Operators -
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