What does the term "boot strap" mean? From time to time or when reading a paper I hear the term "bootstrap" or "the bootstraping technique" or similar terminology. I cannot find a concise reference or explanation as to what is this method since when I google it I find all kinds of different things that are named as such. Is there a unifying philosophy that binds all these methods, techiniques, tricks, etc? 
 A: As is surely visible in many (older?) sources, the original colloquial sense of this was a partly ironic admonition to "lift yourself up by your own boot-straps" (bootstraps being loops or flaps at the sides of the top of boots that you'd pull on to help get your boots on). Of course, one cannot lift oneself by one's own bootstraps. I do not know how sarcastic or facetious this usage was. (I'm thinking of U.S. English, perhaps back to UK English.)
In various mathematical situations, what it really amounts to is induction, which can reasonably be portrayed as a semi-magical bootstrapping, after all, despite the fact that this is not possible "in the real world". For example, if $\Delta u = \lambda u$ with non-zero $\lambda$ and the Laplacian $\Delta$ on a manifold (e.g., the real line), and we know that $u$ is in some Sobolev space $H^s$, then since we also know that $\Delta$ maps $H^s$ to $H^{s-2}$ (for all $s$), solving such an equation maps $H^s$ to $H^{s+2}$. An induction shows that $u$ is in the nested intersection (projective limit) of the $H^s$'s, which, by Sobolev imbedding, consists of smooth functions. This is a way to prove a particular (important) instance of "elliptic regularity", for example.
