I am very confused as nothing has been mentioned in the question -the metric .Howevere I do have an idea about compactness. Somebody hints (not the full answer) will be appreciated . I will try to figure it out.
2026-03-25 16:01:22.1774454482
Is $GL_n(\Bbb(R))$ and $SL_n(\Bbb (R)) $ compact?
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Assuming that we see them as subspaces of the space of all real $n\times n$ matrices, with any norm whatsoever (for instance, $\|(a_{ij})_{1\leqslant i,j\leqslant n}\|=\sqrt{\sum_{i,j=1}^na_{ij}^{\,2}}$, but it doesn't really matter, since all norms on a finite-dimensional real vector space are equivalent), then neither of them is compact, since both of them are unbounded: consider the matrices of the form $\operatorname{diag}\left(n,\frac1n,1,1\ldots,1\right)$.