Timeline for Banach-like analysis on metric spaces
Current License: CC BY-SA 4.0
5 events
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| Jul 27, 2021 at 5:55 | comment | added | Onur Oktay | @WlodAA except it embeds the metric space $X$ in a larger metric space with an algebraic (vector space) structure. If $X$ already possess some other algebraic structure, for example if $X$ is Lie group, then the embedding above would not preserve this extra structure. In that case, perhaps 1-parameter subgroups & continuous group homomorphisms are what one might want to study. Various theorems of functional analysis has analogous versions for continuous homomorphisms on Lie groups. | |
| Jul 27, 2021 at 5:15 | comment | added | Wlod AA | @OnurOktay, this is a very well-known embedding but I don't think that this one does anything algebraic to the given metric space. | |
| Jul 27, 2021 at 0:29 | comment | added | Onur Oktay | Generally metric spaces do not possess such segments as you mention, unless you assume some extra structure. Thus, the embedding above provides you a way to define linear segments. On the other side, if there is a manifold structure (take torus as an example) on your metric space, you may choose to work with geodesics. Geodesics are analogous to linear segments in the way that they minimize the distance between given two points. It might be interesting to study the functions that maps geodesics onto geodesics. | |
| Jul 26, 2021 at 23:21 | comment | added | Kacper Kurowski | While I see that the entirety of a metric structure would be encoded in such an embedding, I'm not sure whether it would preserve the mentioned "segmental" structure in a sense that this embedding might not map "segments" in a metric space to segments in a normed space. Because of that, I worry that such an embedding might obfuscate some structure that might be more apparent if we were to look at the metric space in a more intrinsic way. | |
| Jul 26, 2021 at 22:28 | history | answered | Onur Oktay | CC BY-SA 4.0 |