7

Here's an example of what I mean by "job description" in the post's title:

triangle inequality: to be used, whenever the (unsigned) distances between adjacent points in a sequence $x_0, x_1, x_2, \dots, x_{n>1}$ are given, to derive an upper bound on the (unsigned) distance between the endpoints $x_0$ and $x_n$.1

Is it possible to give a similarly high-level characterization of the kind of problem that Hölder's inequality is useful for?

BTW, the generality of the theorem's statement, which always involves exponents $s, t \in [1, \infty]$ satisfying $s^{-1}+t^{-1} = 1$, greatly exceeds the modest needs of every application area I ever come across, where only the case $s = t = 2$ ever matters. This may have something to do with my inability to formulate a job description for this inequality.

1This is the basic idea, though there are several variations of it. Also, I don't claim that this is, even "in essence", the only way to interpret the triangle inequality's utility. This "job description" is only meant to make concrete the difference between knowledge that is somehow "in active use", and that, like my present "knowledge" of the Hölder's inequality, which is passive at best.

kjo
  • 14,334

2 Answers2

1

We might say the following about Holder's inequality:

Holder's inequality bounds the integrability of the product of two functions, the first whose $p$th power is integrable, and the second whose $q$th power is integrable, by the respective $p$- and $q$-norms of those functions. More broadly, Holder's inequality tells us that given a function whose $p$th power is integrable, the product of $f$ with every function in a space isomorphic to the dual of the space to which $f$ belongs is an absolutely integrable function.

In other words, $\|fg\|_1 \le \|f\|_p\|g\|_q$ for $f \in L^p$, $g \in L^q$.

Restricting to $\ell^p$ and $\ell^q$, we can say

The Manhattan distance of the product of two functions is bounded by the $p$th root of the first ($p$-summable) function and the $q$th root of the second ($q$-summable) function.

If you want to get more broad, we could say the following:

Given a function whose $p$th powers are summable/integrable, and another function whose $q$th powers are summable/integrable, then the unsigned distances/area of the product of those functions must be finite.

Emily
  • 35,688
  • 6
  • 93
  • 141
1

As you said, the triangle inequality lets you use bounds on (norms of) terms to get a bound on (the norm of) their sum. Hölder's inequality lets you use bounds on (integrals of powers of) factors to get a bound on (the integral of) their product. Just as, when using the triangle inequality, sometimes it's not obvious how to split up the thing you want to bound into summands, so with Hölder, often the tricky bit is to figure out how best to split your integrand up into factors.

It might be useful to look at some examples of applications of Hölder with this notion of splitting-into-factors in mind. So, examples: Liapounov's inequality; Littlewood's inequality; Young's convolution inequality; log-concavity of binomial coefficients.

(Incidentally, sometimes when bounding a product it's enough to use AM/GM, possibly with weights. Hölder is what you get when you combine AM/GM with homogeneity.)

Community
  • 1