Normed Vector Spaces I
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### Lecture Description

Lecture with Ole Christensen. Kapitler: 00:00 - Introduction; 06:45 - Vector Spaces; 07:15 - Example 1; 12:00 - Mathematical Tool - Fourier Transform; 17:00 - Example 2; 20:00 - Example 3; 23:00 - New Concept - Norm; 27:45 - Lemma 2.1.2 - The Opposite Triangle Inequality; 35:15 - Convergence; 39:30 - Exact Definition; 41:45 - Goal; 43:15 - Subspaces; 44:15 - Characterisation Of A Subspace; 46:15 - Example: A Trigonometric Polynomial;

### Course Description

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.