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# Mathematics
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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".
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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
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5.4 Continuiuty
5.5 Convergence of Function Sequences
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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
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7.4 Product Measures and the Fubini Theorem
7.5 The Transformation Theorem
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7.6 Lebesgue-Stieltjes Integral
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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)
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11.3 Outlook: Tempered Distributions
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12. Operator Theory
12.1 Linear Operators
12.2 Dual Spaces
12.3 Hilbert Spaces
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12.4 Adjoint Operators
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13. Spectral Theory
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13.1 Spectral Theory of Bounded Linear Operators
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13.2 Spectral Representation of Bounded Self-Adjoint Operators
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13.3 Compact & Unbounded Linear Operators