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Lecture V - Topological Groups

in A topological group is a topological space which is also a group such that the group operations (multiplication and inversion) are continuous. They arise naturally as continuous groups of symmetries of topological spaces. A case in point is the group $SO(3, \mathbb{R})$ of rotations of $\mathbb{R}^3$ about the origin which is a group of symmetries of the sphere $S^2$. Many familiar examples of topological spaces are in fact topological groups. The most basic example of-course is the real line with the group structure given by addition. Other obvious examples are $\mathbb{R}^n$ under addition, the multiplicative group of unit complex numbers $S^1$ and the multiplicative group $\mathbb{C}^{*}$.

In the previous lectures we have seen that the group $SO(n, \mathbb{R})$ of orthogonal matrices with determinant one and the group $U(n)$ of unitary matrices are compact. In this lecture we initiate a systematic study of topological groups and take a closer look at some of the matrix groups such as $SO(n, \mathbb{R})$ and the unitary groups $U(n)$.