Tricks Wiki article: The tensor power trick

As many readers may already know, my good friend and fellow mathematical blogger Tim Gowers, having wrapped up work on the Princeton Companion to Mathematics (which I believe is now in press), has begun another mathematical initiative, namely a “Tricks Wiki” to act as a repository for mathematical tricks and techniques.    Tim has already started the ball rolling with several seed articles on his own blog, and asked me to also contribute some articles.  (As I understand it, these articles will be migrated to the Wiki in a few months, once it is fully set up, and then they will evolve with edits and contributions by anyone who wishes to pitch in, in the spirit of Wikipedia; in particular, articles are not intended to be permanently authored or signed by any single contributor.)

So today I’d like to start by extracting some material from an old post of mine on “Amplification, arbitrage, and the tensor power trick” (as well as from some of the comments), and converting it to the Tricks Wiki format, while also taking the opportunity to add a few more examples.

Title: The tensor power trick

Quick description: If one wants to prove an inequality for some non-negative quantities X, Y, but can only see how to prove a quasi-inequality that loses a multiplicative constant C, then try to replace all objects involved in the problem by “tensor powers” of themselves and apply the quasi-inequality to those powers.  If all goes well, one can show that for all , with a constant C which is independent of M, which implies that as desired by taking roots and then taking limits as .