Lecture with Ole Christensen. Kapitler: 00:00 - Intro To Approximation Theory; 10:00 - Remarks On Vectorspaces In Mat4; 13:30 - Def.: Dense Subset; 19:15 - Dense Subspace Of The Sequence Spaces L^p; 24:45 - Dense Subspace Of The Function Spaces L^p; 35:15 - Weierstrass Approximation Theorem;

1. Normed Vector Spaces I
2. Normed Vector Spaces II
3. Banach Spaces I
4. Banach Spaces II
5. Hilbert Spaces I
6. Hilbert Spaces II
7. Adjoint Operator I
8. Adjoint Operator II
9. Lp Spaces on the real line
10. Lp Spaces On The Real Line II
11. More On Lp And L2 Spaces I
12. More On Lp And L2 Spaces II
13. More On Operators On L2 I
14. Orthonormal Bases Vs Fourier Series II
15. Approximation Theory I
16. Approximation Theory II
17. The Fourier Transform I
18. The Fourier Transform II
19. Fourier Transform And Wavelets I
20. The Fourier Transform And Wavelets II
21. Wavelets And Multiresolution Analysis I
22. Wavelets And Multiresolution Analysis II
23. Wavelets And B-Splines I
24. Wavelets And B-Splines II
25. Special Functions And Diff. Equation Course Evaluation

This is a Master's graduate-level course on real analysis. A student who has met the objectives of the course will be able to:
- distinguish between normed spaces and Hilbert spaces
- understand various types of convergence and how to verify them
- master basic operations in Hilbert spaces
- understand the role of linear algebra in analysis
- know the role of L^2 and perform basic operations herein
- master the basic manipulations with Fourier transform
- know when one should apply Fourier series or the Fourier transform
- expand square-integrable functions in various bases
- Perform calculations on B-splines
- Perform calculations with the L^p-spaces and the corresponding sequence spaces
- master basic wavelet theory

Some of the topics covered include: Normed vector spaces, Hilbert spaces, bases in Hilbert spaces, basic operator theory, the spaces L^p and l^p, approximation, the Fourier transform, convolution, the sampling theorem, B-splines, special basis functions (e.g, Legendre and Hermite polynomials), an introduction to wavelet theory.