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1. • CommentRowNumber1.
   • CommentAuthorAndrew Stacey
   • CommentTimeNov 29th 2011
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   Author: Andrew Stacey
   Format: MarkdownItexFollowing a discussion on the algebraic topology list, I've written a proof of the contractibility of the space of embeddings of a smooth manifold in a reasonably arbitrary locally convex topological vector space. The details are on [[embedding of smooth manifolds]] and it also led to me creating [[shift space]] (I [checked on MO](http://mathoverflow.net/questions/82108/is-there-a-standard-notation-for-a-shift-space-in-functional-analysis) to see if there was an existing name for this, and Bill Johnson said he hadn't heard of it).

   Following a discussion on the algebraic topology list, I’ve written a proof of the contractibility of the space of embeddings of a smooth manifold in a reasonably arbitrary locally convex topological vector space. The details are on embedding of smooth manifolds and it also led to me creating shift space (I checked on MO to see if there was an existing name for this, and Bill Johnson said he hadn’t heard of it).

2. • CommentRowNumber2.
   • CommentAuthorAndrew Stacey
   • CommentTimeNov 29th 2011
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   Author: Andrew Stacey
   Format: MarkdownItexAdded the standard proof of the contractibility of the infinite sphere to [[sphere]]. Checking the usenet link, I see that this is the proof given there. It's also the proof I gave on MO some time ago. I've no idea who originally came up with it - I suspect it's been reinvented many, many times.

   Added the standard proof of the contractibility of the infinite sphere to sphere. Checking the usenet link, I see that this is the proof given there. It’s also the proof I gave on MO some time ago. I’ve no idea who originally came up with it - I suspect it’s been reinvented many, many times.

3. • CommentRowNumber3.
   • CommentAuthorjim_stasheff
   • CommentTimeNov 29th 2011
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   Author: jim_stasheff
   Format: TextThe oldest example of which I'm aware for the V iso V + V trick is in proving that O(n) is homotopy commutative, so to speak, within O(2n). For V = R^infty and V iso R + V, the oldest is Barry Mazur's proof of Schoedfle's theorem using n-stocks.

   The oldest example of which I'm aware for the V iso V + V trick is in proving that O(n) is homotopy commutative,
   so to speak, within O(2n). For V = R^infty and V iso R + V, the oldest is Barry Mazur's proof of Schoedfle's theorem
   using n-stocks.
4. • CommentRowNumber4.
   • CommentAuthorAndrew Stacey
   • CommentTimeNov 30th 2011
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   Author: Andrew Stacey
   Format: MarkdownItexTom Goodwillie wanted an embedding with a closed image. I've added that to the page, under certain assumptions on the background space. I'm pretty sure that those assumptions are too strong, since they don't work for $\ell^2$, but it might be that to get something that works for $\ell^2$ then we need to be a bit more careful with the embedding. The key is showing that if a net $(x_\lambda)$ contained in the image converges in the ambient space then its limit, say $x$, is non-zero and that non-zeroness is detectable by something that looks like one of the partition of unity functions (ie when composed with $\psi$ it becomes one of the partition of unity functions). So we need to ensure that the image of the embedding is bounded away from $0$, and this "detectability of limits" property holds. I need to think a bit more about when these happen. (Incidentally, any functional analysts here who want to say something sensible at [[equicontinuity]] are welcome to do so. Otherwise, I'll "copy"[1] a load of stuff out of Schaefer's book.) [1] not literally, of course.

   Tom Goodwillie wanted an embedding with a closed image. I’ve added that to the page, under certain assumptions on the background space. I’m pretty sure that those assumptions are too strong, since they don’t work for $\ell^2$, but it might be that to get something that works for $\ell^2$ then we need to be a bit more careful with the embedding.

   The key is showing that if a net $(x_\lambda)$ contained in the image converges in the ambient space then its limit, say $x$, is non-zero and that non-zeroness is detectable by something that looks like one of the partition of unity functions (ie when composed with $\psi$ it becomes one of the partition of unity functions). So we need to ensure that the image of the embedding is bounded away from $0$, and this “detectability of limits” property holds.

   I need to think a bit more about when these happen.

   (Incidentally, any functional analysts here who want to say something sensible at equicontinuity are welcome to do so. Otherwise, I’ll “copy”[1] a load of stuff out of Schaefer’s book.)

   [1] not literally, of course.

5. • CommentRowNumber5.
   • CommentAuthorAndrew Stacey
   • CommentTimeNov 30th 2011
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   Author: Andrew Stacey
   Format: MarkdownItexOh, and: Jim, thanks for the references. I know of the first, and I agree it's a nice example. I don't know what the second refers to and my search skills aren't enough for me to find it. Did you spell the theorem name correctly?

   Oh, and: Jim, thanks for the references. I know of the first, and I agree it’s a nice example. I don’t know what the second refers to and my search skills aren’t enough for me to find it. Did you spell the theorem name correctly?

6. • CommentRowNumber6.
   • CommentAuthorAndrew Stacey
   • CommentTimeNov 30th 2011
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   Author: Andrew Stacey
   Format: MarkdownItexI figured out a better way of ensuring that one had an embedding with closed image. The key was to look at the projection functionals rather than the coordinate vectors. There's a nice geometrical picture of how this works: as one goes into higher dimensions, one "spreads out" the embedding to ensure that nothing converges that shouldn't. In searching for equicontinuity, I came up with [[subsets of lctvs]]. I guess it makes sense to redirect to there and add a definition there, at least for now. I remember the point of that being that it might be easier to understand all these different types of subset if the definitions are in one place. (That then led me to [[DF space]] where I appear to have had a little fun with the CSS!)

   I figured out a better way of ensuring that one had an embedding with closed image. The key was to look at the projection functionals rather than the coordinate vectors. There’s a nice geometrical picture of how this works: as one goes into higher dimensions, one “spreads out” the embedding to ensure that nothing converges that shouldn’t.

   In searching for equicontinuity, I came up with subsets of lctvs. I guess it makes sense to redirect to there and add a definition there, at least for now. I remember the point of that being that it might be easier to understand all these different types of subset if the definitions are in one place.

   (That then led me to DF space where I appear to have had a little fun with the CSS!)