Talk:Kuratowski embedding

Why not just bounded contiuous functions
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)

Add specificity to the reference to the Frechet paper
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 $$\epsilon_p=\inf_kd(A_p,A_k)$$ 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