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What does it mean when people say that a group is a bounded subset of another group?

I need decide whether $SO_2(\mathbb C)$ is a bounded subset of $\mathbb C^{2\times 2}$. What definition should I use? What is the "distance" function?

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  • $\begingroup$ $SO_2(\mathbb C)$ is certainly closed (as a set of solutions of continuous equations). Thus, it is bounded iff it is compact, for any metric compatible with the standard topology on $\mathbb C^{2\times 2}$. $\endgroup$ – lisyarus May 7 at 21:57
  • $\begingroup$ @lisyarus What are the definitions of closedness and compactness of a group? What is the definition of that metric? $\endgroup$ – user398843 May 7 at 22:02
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Take any norm on $\mathbb C^{2\times2}$ (for instance $\left\lVert\left[\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right]\right\rVert=\sqrt{\lvert a\rvert^2+\lvert b\rvert^2+\lvert c\rvert^2+\lvert d\rvert^2}$) and use the distance $d(M,N)=\lVert M-N\rVert$.

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