MAT4 (Ole Christensen) Review

The core concept of this class is approximation. The class provides the fundamental mathematical tools for approximation. Suppose that we have a function $$f \in L^2({\mathbb R})$$ where $$L^2(\mathbb R)$$ is a ][Special vector spaces|fairly generic space]] that covers a broad range of functions. The course uses the concept of denseness to show that any function in $$L^(\mathbb R)$$ can be approximated with a continuous function $$g$$ that has compact support (i.e., $$g \in C_c(\mathbb{R})$$). This statement requires the introduction of core concepts such as Banach spaces], [[Subset closure, Norm, Operator norm, Bounded linear operator etc.

The course starts from abstract and goes toward more concrete and applied concepts. Specifically, it starts with the more general topics of convergence in normed spaces, inner vector spaces, Banach spaces Hilbert spaces, bounded linear operators on Banach/Hilbert spaces etc. It then goes to specific Banach/Hilbert spaces such as $$L^2(\mathbb R), L^1(\mathbb R), C_c(\mathbb R), C_0(\mathbb R)$$. The concept of asis in finite and infinite spaces is introduced. The course defines specific critical Bounded linear operations on $$L^2(\mathbb R)$$ such as the Fourier transform or more basic operators such as translation operator, modulation operator and dilation operator. All those operators then lead to the very important Wavelet theory, which is of great utility for real life applications.

The definitions and theorems etc. referred to below are from the book.

Week 1

 * Introduces norm
 * Motivates usage of normed vector spoces
 * Proves negative triangle inequality
 * Definition of sequence convergence (Def. 2.1.5.)

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