Somewhat surprisingly, Jean Bourgain's 1985 theorem about embedding finite metric spaces into Hilbert spaces was not mentioned yet. Since metrics have not mentioned at all in this thread, let me mention this example now. 

The topic of representing *prescribed*, possibly **strange** finite metric spaces 'as best one can' by **non-strange**, traditional metric spaces is important e.g. in **computational biology**. Very briefly, I think the essence of all of this is  

> **not** to have to **store** a table of pairwise distances among any two of a large number of proteins, which after all would take space *quadratic* in the number of 'specimens', rather *label*/*represent* each protein by an element of a traditional metric space, and then **calculate** a distance the traditional way, from the traditional representatives, **on an as-needed-basis**. 

For this to result in only a small error of approximation (of the 'true' distance w.r.t. some new-fangled 'similarity distance' between proteins), one needs to know how well such 'representations'/'low-distortion embeddings' can be done in principle. 

Bougain used the "probabilistic method" to prove a *lower bound* on how well this can be done.   

 The publication is [Jean Bourgain: *On lipschitz embedding of finite metric spaces in Hilbert space*. Israel Journal of Mathematics
March 1985, Volume 52, Issue 1–2, pp 46–52](https://link.springer.com/article/10.1007/BF02776078), and the abstract reads:

> "It is shown than any $n$ point metric space is up to $\log n$ [lipeomorphic](https://math.stackexchange.com/questions/1445936/what-is-preserved-by-lipeomorphisms/1448584) to a subset of Hilbert space. We also exhibit an example of an $n$-point metric space which cannot be embedded in Hilbert space with distortion less than $(\log n)/(\log\log n)$, showing that the positive result is essentially best possible. **The methods used are of probabilistic nature. For instance, to  construct our example, we make use of random graphs.**"

(emphasis added)