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Keep parameters unspecified until it is clear how to optimize them

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Quick description

In the course of proving some statement, you may want to introduce some parameter (e.g. if you wish to divide into cases when a certain variable x is small or large, one may introduce a parameter R and divide into the cases |x|<R and |x| \geq R.) But you don't yet know what the "best" choice of this new parameter is. In many cases, what one can do is just leave the parameter unspecified, and continue the argument with this undetermined parameter. At a much later stage of the argument, it may become clearer what metric one would use to decide what choices of parameter are "good" and what are "bad", at which point one can optimize in that metric and find the right choice of parameter.

Prerequisites

Example 1

Suppose one wishes to establish the Cauchy-Schwarz inequality

 \sum_{n=1}^N a_n b_n \leq \| a \| \|b\|

for non-negative real numbers a_1,\ldots,a_N,b_1,\ldots,b_N, where

= (\sum_{n=1}^N |a_n|^2)^{1/2}; \quad \|b\| := (\sum_{n=1}^N |b_n|^2)^{1/2}.

A first attempt would be to use the arithmetic mean-geometric mean inequality

 a_n b_n \leq \frac{1}{2} a_n^2 + \frac{1}{2} b_n^2(1)

but when one substitutes this in and works everything out, one gets the inequality

 \sum_{n=1}^N a_n b_n \leq \frac{1}{2} \| a \|^2 + \frac{1}{2} \|b\|^2.

which is inferior to Cauchy-Schwarz (by another application of the AM-GM inequality!).

However, one can be cleverer about this by multiplying a_n by an unspecified parameter \lambda > 0 and dividing b_n by the same parameter, which when inserted into (1) gives the more general inequality

 a_n b_n \leq \frac{\lambda^2}{2} a_n^2 + \frac{1}{2\lambda^2} b_n^2.(2)

It is not yet clear what value of \lambda is best (though we already know that \lambda=1 does not always work). But one can forge on regardless. Summing (2) in n, one obtains

 \sum_{n=1}^N a_n b_n \leq \frac{\lambda^2}{2} \|a\|^2 + \frac{1}{2\lambda^2} \|b\|^2.

And now it is clear what to do to optimize in \lambda: one should choose \lambda so that the right-hand side is as small as possible. A little calculus (see also the heuristic "To optimize a sum, try making the terms roughly equal in size") then shows that the optimal choice of \lambda is \|b\|^{1/2}/\|a\|^{1/2}, at least when \|a\| and \|b\| are both non-zero (but the case when \|a\|=0 or \|b\|=0 can be easily handled separately). Inserting this choice of \lambda gives the desired inequality.

General discussion

Comments

Amusingly, I planned an

Amusingly, I planned an article ages ago that was very similar in spirit to this one. I've just looked for a dead link but can't find one. Anyhow, it's great that you've started this one, which I hope may eventually fit in nicely as a special case of a more general principle, which is what my intended article was going to be about. The general principle is that whenever you need to choose something and it feels difficult to do so because you need to guess too much about how the rest of the proof will go, you have the option of not choosing it and instead taking some abstract object instead, running the rest of the proof, and seeing what properties you need your unspecified object to have. At that point, you have a new problem: is there an object with those properties? Often this problem is much clearer and easier than the problem you had before when you didn't know how the proof would go.

Numerical examples in analysis are an obvious source of examples. I use the general idea a lot, but when it came to writing the article I suddenly found I couldn't think of any. Perhaps I'll write a stub with a general discussion – roughly equal to the above paragraph – but no examples! And then it could be a parent for this article.