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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMay 22nd 2017
• (edited May 22nd 2017)

at general linear group we only had some sentences on its incarnation as an algebraic group. I have started a subsection Definition – As a topological group with some basics.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMay 29th 2017

What’s a quick proof that the topology on $GL(n,k)$ as a subspace of $\mathbb{R}^{n^2}$ with its Euclidean topology coincides with that as a subspace of $Maps(k^n,k^n)$ with its compact-open topology?

• CommentRowNumber3.
• CommentAuthorTodd_Trimble
• CommentTimeMay 29th 2017

I guess $k = \mathbb{R}$?

So we do have an injective continuous map $GL(n, \mathbb{R}) \to Map(\mathbb{R}^n, \mathbb{R}^n)$, the currying of the continuous action $GL(n, \mathbb{R}) \times \mathbb{R}^n \to \mathbb{R}^n$. This says that the usual Euclidean topology is finer than the subspace topology coming from $Map(\mathbb{R}^n, \mathbb{R}^n)$.

But the Euclidean topology is also coarser. Let’s do this for $M(n, \mathbb{R})$ instead of $GL(n, \mathbb{R})$. A Euclidean neighborhood base of a linear map or matrix $A$ consists of sets of the form $\{B: \forall_{1 \leq i \leq n}\; |A e_i - B e_i| \lt \epsilon\}$. But this is a basis/base element for the function space topology, $\bigcap_{i = 1}^n C(K_i, U_i)$ where $K_i = \{e_i\}$ and $U_i$ is the $\epsilon$-ball about $A e_i$.

(It may help to think of convergence in the function space topology as the exact same as uniform convergence over every compact set.)

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeMay 29th 2017

Thanks, Todd!

I have added that to the entry here.