Talk:Kuratowski embedding

Why not just bounded contiuous functions[edit]

Hi, the article seems to me very clear and well written; however I took the liberty of changing a bit the presentation of the two embedding theorems, making the embedding directely in the Banach space $\scriptstyle C_{b}(X)$ . The second form that was mentioned, i.e. the space of bounded functions as target, does not seem so relevant, and it is just a consequence of the inclusion of $\scriptstyle C_{b}(X)$ into the space of bounded functions. But if you have a special reason to consider also the larger space I apologize. In any case I would avoid the notation $\scriptstyle \ell ^{\infty }(X)$ , since $\ell$ is used preferably for spaces of sequences (so this one would better used to denote the space of bounded sequences in X; for the space of bounded functions on X then $\scriptstyle {\mathcal {B}}(X)$ is somehow more standard).--PMajer (talk) 17:42, 13 January 2009 (UTC)[reply]

Add specificity to the reference to the Frechet paper[edit]

Went through the entire 72-page paper at least three times, both reading and searching. Could not find any reference to such an embedding -- the closest instance I could find is that in Item 51, part 2 of the theorem, he defines $\epsilon _{p}=\inf _{k}d(A_{p},A_{k})$ for $A_{k}\in E$ a metric space. He doesn't do anything remotely like an embedding with it, though -- rather, he uses it to find a family of disjoint sets which he does some further constructions with. The other two points that look relevant are items 62 and 68, defining a metrization of pointwise convergence for sequences and defining the $\ell ^{\infty }$ norm, but again there's no motion towards an embedding.

On the other hand, I see references explicitly attributing the construction to Fréchet, so I'm probably missing something. Examples:

J. Matoušek. Lectures on discrete geometry. Springer-Verlag, New York, 2002, Ch1, p. 17
Bartal, Yair, et al. “Limitations to Fréchet’s Metric Embedding Method.” Journal of Mathematics, vol. 151, no. 1, Dec. 2006, pp. 111–24. Crossref, https://doi.org/10.1007/bf02777357.

However, these cite each other and crucially don't cite Fréchet explicitly. Neither do I see the paper referenced cited anywhere, but that may be due to too-low effort in this particular point.

Also raised this question on MathOverflow, hopefully they'll have an answer https://mathoverflow.net/questions/464735/reference-request-fr%C3%A9chet-embedding