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    • 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)GL(n,k) as a subspace of n 2\mathbb{R}^{n^2} with its Euclidean topology coincides with that as a subspace of Maps(k n,k n)Maps(k^n,k^n) with its compact-open topology?

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 29th 2017

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

    So we do have an injective continuous map GL(n,)Map( n, n)GL(n, \mathbb{R}) \to Map(\mathbb{R}^n, \mathbb{R}^n), the currying of the continuous action GL(n,)× n nGL(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( n, n)Map(\mathbb{R}^n, \mathbb{R}^n).

    But the Euclidean topology is also coarser. Let’s do this for M(n,)M(n, \mathbb{R}) instead of GL(n,)GL(n, \mathbb{R}). A Euclidean neighborhood base of a linear map or matrix AA consists of sets of the form {B: 1in|Ae iBe i|<ε}\{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, i=1 nC(K i,U i)\bigcap_{i = 1}^n C(K_i, U_i) where K i={e i}K_i = \{e_i\} and U iU_i is the ε\epsilon-ball about Ae iA 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.

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