1. Metric spaces
2. Normed spaces
3. Examples
4. Compactness
5. Cardinality of Sets
6. The Stone-Weierstraß theorem
7. Banach‘s fixed point theorem
8. Lp spaces
9. Equivalent norms
10. Compactness in normed spaces
    
    

Kapitel 2. Linear and continuous operators

1. Basics
2. Examples
   
   

Kapitel 3. Main Theorems about Operators

1. Baire‘s theorem
2. Uniform boundedness principle
3. Open mapping theorem
4. Continuous inverse theorem
5. Closed Graph theorem
   
   

Kapitel 4. Hilbert spaces

1. Pre-Hilbert spaces
2. Hilbert spaces and normed spaces
3. Best approximation
4. Projection theorem
5. Fréchet-Riesz representation theorem
6. Orthonormal systems and bases in Hilbert spaces
7. Fischer-Riesz theorem
8. The spectral theorem for compact self-adjoint operators
   
   

Kapitel 5. Dual spaces

1. Examples
2. The Hahn-Banach theorem and its consequences
   
   

Kapitel 6. Spectrum of compact operators – Fredholm theory

1. Adjoint operators
2. The spectrum of bounded operators
3. Fredholm theory