1

Where exactly do $L^p$ norms and $L^p$ spaces show up naturally? In other words, how would you arrive at these concepts out of necessity of solving some other problem in a way that motivates using them?

My attempt: Bernkopf's history points out that they first arose as part of the moment problem:

enter image description here

Kline's Mathematical Thought III says that Riesz was "obliged" to introduce these ideas because he (randomly???!) used Holder and Minkowski's inequalities:

enter image description here

So, taking a simple case as though you were living in 1910 trying to invent functional analysis:

Given a function (a random variable, but thinking heuristically here) $g(x)$ with Taylor (orthogonal) expansion

$$ g(x) = \sum_{n=0}^{\infty} c_n x^n = \sum_{n=0}^\infty c_n g_n(x)$$

and a probability distribution $f(x)$ we know the expected value of $f$ over $I = [0,1]$ is $$\int_I g(x)f(x)dx = \int_I \sum_{i=0}^{\infty} c_n x^n f(x) dx = \sum_{i=0}^{\infty} c_n \int_I x^n f(x) dx = \sum_{i=0}^\infty c_n a_n$$

I can see the necessity, if you want to invert this process (i.e. for a given $g_p(x) = x^p$, finding a probability distribution $f$ such that $\int x^p f(x)dx = a_p$), of defining an $L^p$ space as the space of all functions $h$ whose $p$'th power is integrable with respect to the density $f(x)$: $$L^p = \{ h | \int |h(x)|^p f(x) dx < \infty \}$$ so that for our example $h(x) = x \in L^p$ gives $\int_I x^p f(x) dx < \infty$, but since we will also have $\int_I x^n f(x)dx = \int_I g_n(x) f(x)dx$ terms we also want to have $\int_I g_n(x)^p f(x)dx < \infty$.

But I can't see why, from this, you'd define the $L^p$ norm of $x$ to be

$$||x|| = ( \int_I x^p f(x) dx)^{1/p} $$

or how this gives "the $p$'th moments of a random variable" when what I did above gives them? What is the point of those complicated roots? In other words I don't see why you'd think to invoke Holder or Minkowski from thin air.

My guess is that you would only do this because you knew in advance you would need to bound the integral $\int_I g(x) f(x)dx$ as part of proving some $f$ existed, so you just wanted any inequality you could find, and Holder/Minkowski were just the ones Riesz just knew of?

Can anybody clean this up or make a better version, preferably indicating why you'd know to apply this material to differential equations too?

Community
  • 1
bolbteppa
  • 4,589
  • $L^1$ is natural; it describes mass and similar such things. $L^2$ is essentially natural; it generalizes Euclidean space more or less directly, and there are other obvious reasons to use it, such as aspects of PDE and harmonic analysis. $L^\infty$ is natural, we like to talk about uniform distances. I rarely see the others "jump out from the math". One place is in Sobolev embedding of $W^{k,p}$ where $p=1,2$ or $\infty$. Another place is the Berry-Esseen theorem. – Ian May 11 '16 at 21:37
  • 1
    you might find this useful http://math.stackexchange.com/questions/1683399/why-define-norm-in-l-p-in-that-way/1692234#1692234 – SiXUlm May 11 '16 at 21:40
  • 2
    Riesz knew about Minkowski and H"{o}lder inequalities, and he restricted his attention to cases where these applied, which is why he left out $L^1$. The condition $\int |f|^p < \infty$ was a natural generalization from $L^2$. With that he was led to consider $(\int |f|^p)^{1/p}$, and to use these inequalities to show you end up with a norm. It appears that the decision to consider $\int |f|^p < \infty$ led to the use of Minkowski and H"{o}lder inequalities, and that led to norms. In other words, norms didn't come first, the finite integral conditions did. – Disintegrating By Parts May 12 '16 at 02:10
  • 1
    Was just checking to see how much time I have left – Ryan Unger Apr 25 '19 at 19:05

0 Answers0